### Nuprl Lemma : extend-name-morph-face-map

`∀I,K:Cname List. ∀f:name-morph(I;K). ∀z,x:Cname. ∀i:ℕ2.`
`  (f[z:=x] o (x:=i)) = ((z:=i) o f) ∈ name-morph([z / I];K) supposing (¬(x ∈ K)) ∧ (¬(z ∈ I))`

Proof

Definitions occuring in Statement :  name-comp: `(f o g)` face-map: `(x:=i)` extend-name-morph: `f[z1:=z2]` name-morph: `name-morph(I;J)` coordinate_name: `Cname` l_member: `(x ∈ l)` cons: `[a / b]` list: `T List` int_seg: `{i..j-}` uimplies: `b supposing a` all: `∀x:A. B[x]` not: `¬A` and: `P ∧ Q` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uimplies: `b supposing a` and: `P ∧ Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` not: `¬A` implies: `P `` Q` prop: `ℙ` false: `False` subtype_rel: `A ⊆r B` name-morph: `name-morph(I;J)` face-map: `(x:=i)` name-comp: `(f o g)` extend-name-morph: `f[z1:=z2]` compose: `f o g` uext: `uext(g)` nameset: `nameset(L)` coordinate_name: `Cname` int_upper: `{i...}` nequal: `a ≠ b ∈ T ` squash: `↓T` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` top: `Top` le: `A ≤ B` less_than: `a < b` decidable: `Dec(P)` sq_stable: `SqStable(P)` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` isname: `isname(z)` true: `True`
Lemmas referenced :  name-morphs-equal cons_wf coordinate_name_wf name-comp_wf extend-name-morph_wf face-map_wf l_member_wf istype-void int_seg_wf name-morph_wf nameset_wf eq-cname_wf eq_int_wf int_seg_properties full-omega-unsat intformand_wf intformeq_wf itermVar_wf intformnot_wf istype-int int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int int_subtype_base decidable__equal_int istype-le assert-eq-cname bool_wf iff_weakening_uiff assert_wf equal-wf-T-base set_subtype_base le_wf iff_imp_equal_bool le_int_wf btrue_wf iff_functionality_wrt_iff true_wf assert_of_le_int iff_weakening_equal istype-true bfalse_wf false_wf intformle_wf itermConstant_wf intformless_wf int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_less_lemma nsub2_subtype_extd-nameset isname-nameset cons_member isname_wf assert-isname extd-nameset_subtype_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt isect_memberFormation_alt cut sqequalHypSubstitution productElimination thin introduction extract_by_obid isectElimination hypothesis hypothesisEquality independent_isectElimination sqequalRule productIsType functionIsType universeIsType natural_numberEquality inhabitedIsType because_Cache applyEquality lambdaEquality_alt setElimination rename functionExtensionality applyLambdaEquality imageMemberEquality baseClosed imageElimination independent_functionElimination voidElimination approximateComputation dependent_pairFormation_alt int_eqEquality dependent_functionElimination Error :memTop,  independent_pairFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry equalityIstype promote_hyp instantiate isect_memberEquality_alt cumulativity intEquality dependent_set_memberEquality_alt

Latex:
\mforall{}I,K:Cname  List.  \mforall{}f:name-morph(I;K).  \mforall{}z,x:Cname.  \mforall{}i:\mBbbN{}2.
(f[z:=x]  o  (x:=i))  =  ((z:=i)  o  f)  supposing  (\mneg{}(x  \mmember{}  K))  \mwedge{}  (\mneg{}(z  \mmember{}  I))

Date html generated: 2020_05_21-AM-10_49_02
Last ObjectModification: 2019_12_08-PM-07_06_29

Theory : cubical!sets

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