Nuprl Lemma : cat-comp-isomorphism
∀C:SmallCategory. ∀a,b,c:cat-ob(C). ∀f:cat-arrow(C) a b. ∀g:cat-arrow(C) b c.
(cat-isomorphism(C;a;b;f) ⇒ cat-isomorphism(C;b;c;g) ⇒ cat-isomorphism(C;a;c;cat-comp(C) a b c f g))
Proof
Definitions occuring in Statement :
cat-isomorphism: cat-isomorphism(C;x;y;f),
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
cat-isomorphism: cat-isomorphism(C;x;y;f),
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
cat-inverse: fg=1,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cat-isomorphism_wf,
cat-arrow_wf,
cat-ob_wf,
small-category_wf,
cat-comp_wf,
cat-inverse_wf,
equal_wf,
iff_weakening_equal,
cat-comp-ident2,
squash_wf,
true_wf,
cat-comp-assoc,
cat-comp-ident1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
dependent_pairFormation,
independent_pairFormation,
productEquality,
equalityUniverse,
levelHypothesis,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
because_Cache,
lambdaEquality,
imageElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
universeEquality
Latex:
\mforall{}C:SmallCategory. \mforall{}a,b,c:cat-ob(C). \mforall{}f:cat-arrow(C) a b. \mforall{}g:cat-arrow(C) b c.
(cat-isomorphism(C;a;b;f)
{}\mRightarrow{} cat-isomorphism(C;b;c;g)
{}\mRightarrow{} cat-isomorphism(C;a;c;cat-comp(C) a b c f g))
Date html generated:
2020_05_20-AM-07_50_12
Last ObjectModification:
2017_07_28-AM-09_19_04
Theory : small!categories
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