Nuprl Lemma : cat-isomorphic_transitivity
∀C:SmallCategory. ∀a,b,c:cat-ob(C). (cat-isomorphic(C;a;b) ⇒ cat-isomorphic(C;b;c) ⇒ cat-isomorphic(C;a;c))
Proof
Definitions occuring in Statement :
cat-isomorphic: cat-isomorphic(C;x;y),
cat-ob: cat-ob(C),
small-category: SmallCategory,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
cat-isomorphic: cat-isomorphic(C;x;y),
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ
Lemmas referenced :
cat-comp_wf,
cat-comp-isomorphism,
cat-isomorphism_wf,
cat-isomorphic_wf,
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
applyEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}C:SmallCategory. \mforall{}a,b,c:cat-ob(C).
(cat-isomorphic(C;a;b) {}\mRightarrow{} cat-isomorphic(C;b;c) {}\mRightarrow{} cat-isomorphic(C;a;c))
Date html generated:
2020_05_20-AM-07_50_19
Last ObjectModification:
2017_01_08-PM-01_40_04
Theory : small!categories
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