Nuprl Lemma : nat-trans-assoc-comp-equation
∀[C,D:SmallCategory]. ∀[F,G,H:Functor(C;D)]. ∀[t1:nat-trans(C;D;F;H)]. ∀[t2:nat-trans(C;D;H;G)]. ∀[A,B,B':cat-ob(C)].
∀[g:cat-arrow(C) A B]. ∀[h:cat-arrow(C) B B'].
((cat-comp(D) (F A) (G B) (G B')
(cat-comp(D) (F A) (F B) (G B) (F A B g) (cat-comp(D) (F B) (H B) (G B) (t1 B) (t2 B)))
(G B B' h))
= (cat-comp(D) (F A) (F B') (G B') (cat-comp(D) (F A) (F B) (F B') (F A B g) (F B B' h))
(cat-comp(D) (F B') (H B') (G B') (t1 B') (t2 B')))
∈ (cat-arrow(D) (F A) (G B')))
Proof
Definitions occuring in Statement :
nat-trans: nat-trans(C;D;F;G),
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
nat-trans: nat-trans(C;D;F;G),
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
cat-arrow_wf,
functor-ob_wf,
nat-trans-assoc-equation,
trans-comp_wf,
cat-comp_wf,
functor-arrow_wf,
nat-trans_wf,
iff_weakening_equal,
trans_comp_ap_lemma,
cat-ob_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G,H:Functor(C;D)]. \mforall{}[t1:nat-trans(C;D;F;H)]. \mforall{}[t2:nat-trans(C;D;H;G)].
\mforall{}[A,B,B':cat-ob(C)]. \mforall{}[g:cat-arrow(C) A B]. \mforall{}[h:cat-arrow(C) B B'].
((cat-comp(D) (F A) (G B) (G B')
(cat-comp(D) (F A) (F B) (G B) (F A B g) (cat-comp(D) (F B) (H B) (G B) (t1 B) (t2 B)))
(G B B' h))
= (cat-comp(D) (F A) (F B') (G B') (cat-comp(D) (F A) (F B) (F B') (F A B g) (F B B' h))
(cat-comp(D) (F B') (H B') (G B') (t1 B') (t2 B'))))
Date html generated:
2020_05_20-AM-07_51_50
Last ObjectModification:
2017_07_28-AM-09_19_24
Theory : small!categories
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