Nuprl Lemma : nat-trans-assoc-equation
∀[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[T:nat-trans(C;D;F;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) (T 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)) (T 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,
nat-trans: nat-trans(C;D;F;G),
true: True,
label: ...$L... t,
squash: ↓T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
nat-trans-equation,
cat-comp_wf,
cat-arrow_wf,
cat-ob_wf,
nat-trans_wf,
cat-functor_wf,
small-category_wf,
functor-ob_wf,
functor-arrow_wf,
equal_wf,
cat-comp-assoc,
iff_weakening_equal,
squash_wf,
true_wf,
istype-universe,
functor-arrow-comp,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
universeIsType,
because_Cache,
inhabitedIsType,
setElimination,
rename,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
dependent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
instantiate,
universeEquality
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G:Functor(C;D)]. \mforall{}[T:nat-trans(C;D;F;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) (T 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)) (T B')))
Date html generated:
2020_05_20-AM-07_51_21
Last ObjectModification:
2020_01_04-PM-05_22_31
Theory : small!categories
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