### Nuprl Lemma : iota-two-face-maps

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

Proof

Definitions occuring in Statement :  name-comp: `(f o g)` iota: `iota(x)` face-map: `(x:=i)` name-morph: `name-morph(I;J)` cname_deq: `CnameDeq` coordinate_name: `Cname` list-diff: `as-bs` cons: `[a / b]` nil: `[]` list: `T List` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A` and: `P ∧ Q` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` squash: `↓T` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` all: `∀x:A. B[x]` coordinate_name: `Cname` int_upper: `{i...}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` not: `¬A` false: `False` int_seg: `{i..j-}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` sq_type: `SQType(T)` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  not_wf equal_wf coordinate_name_wf int_seg_wf name-comp-assoc list-diff_wf cname_deq_wf cons_wf nil_wf face-map_wf2 iota_wf name-morph_wf subtype_rel_wf squash_wf true_wf list_wf list-diff2 iff_weakening_equal subtype_rel_self iota-face-map name-comp_wf subtype_base_sq list_subtype_base set_subtype_base le_wf int_subtype_base list-diff-cons-single int_seg_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf cons_member member_singleton or_wf l_member_wf list-diff-cons deq-member_wf bool_wf eqtt_to_assert assert-deq-member eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin hypothesis productEquality extract_by_obid isectElimination hypothesisEquality sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry natural_numberEquality applyEquality lambdaEquality imageElimination universeEquality imageMemberEquality baseClosed independent_isectElimination independent_functionElimination applyLambdaEquality lambdaFormation dependent_functionElimination instantiate cumulativity intEquality setElimination rename dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation dependent_set_memberEquality computeAll addLevel orFunctionality promote_hyp unionElimination equalityElimination hyp_replacement

Latex:
\mforall{}[I:Cname  List].  \mforall{}[x,y,z:Cname].  \mforall{}[i,j:\mBbbN{}2].
(((x:=i)  o  (y:=j))  o  iota(z))  =  (iota(z)  o  ((x:=i)  o  (y:=j)))  supposing  (\mneg{}(x  =  z))  \mwedge{}  (\mneg{}(y  =  z))

Date html generated: 2017_10_05-AM-10_08_27
Last ObjectModification: 2017_07_28-AM-11_16_51

Theory : cubical!sets

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