### Nuprl Lemma : funinv-compose

`∀[n:ℕ]. ∀[f,g:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ].  (inv(f o g) = (inv(g) o inv(f)) ∈ {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} )`

Proof

Definitions occuring in Statement :  funinv: `inv(f)` inject: `Inj(A;B;f)` compose: `f o g` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` prop: `ℙ` squash: `↓T` compose: `f o g` inject: `Inj(A;B;f)` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` implies: `P `` Q` and: `P ∧ Q` label: `...\$L... t` guard: `{T}` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)`
Lemmas referenced :  compose-injections int_seg_wf inject_wf nat_wf funinv_wf2 equal_wf squash_wf true_wf istype-universe funinv-property int_seg_properties nat_properties decidable__equal_int full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_wf decidable__le le_wf less_than_wf intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma decidable__lt intformless_wf int_formula_prop_less_lemma subtype_rel_self iff_weakening_equal subtype_base_sq int_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality hypothesis setElimination rename Error :inhabitedIsType,  sqequalRule Error :isect_memberEquality_alt,  axiomEquality Error :setIsType,  Error :functionIsType,  Error :universeIsType,  natural_numberEquality equalityTransitivity equalitySymmetry applyLambdaEquality imageMemberEquality baseClosed imageElimination Error :dependent_set_memberEquality_alt,  Error :functionExtensionality_alt,  dependent_functionElimination applyEquality Error :lambdaEquality_alt,  independent_functionElimination universeEquality productElimination unionElimination independent_isectElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality voidElimination independent_pairFormation Error :productIsType,  instantiate cumulativity intEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f,g:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\}  ].    (inv(f  o  g)  =  (inv(g)  o  inv(f)))

Date html generated: 2019_06_20-PM-01_17_42
Last ObjectModification: 2018_10_07-AM-00_37_00

Theory : int_2

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