Nuprl Lemma : nat-trans-comp-equation2
∀[C,D,E:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[J:Functor(D;E)]. ∀[tFG:nat-trans(C;D;F;G)]. ∀[A,B:cat-ob(C)].
∀[g:cat-arrow(C) A B]. ∀[v:cat-ob(E)].
((cat-comp(E) (ob(J) (ob(F) A)) (ob(J) (ob(G) B)) v
(cat-comp(E) (ob(J) (ob(F) A)) (ob(J) (ob(G) A)) (ob(J) (ob(G) B)) (arrow(J) (ob(F) A) (ob(G) A) (tFG A))
(arrow(J) (ob(G) A) (ob(G) B) (arrow(G) A B g))))
= (cat-comp(E) (ob(J) (ob(F) A)) (ob(J) (ob(G) B)) v
(cat-comp(E) (ob(J) (ob(F) A)) (ob(J) (ob(F) B)) (ob(J) (ob(G) B)) (arrow(J) (ob(F) A) (ob(F) B) (arrow(F) A B g))
(arrow(J) (ob(F) B) (ob(G) B) (tFG B))))
∈ ((cat-arrow(E) (ob(J) (ob(G) B)) v) ⟶ (cat-arrow(E) (ob(J) (ob(F) A)) v)))
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,
function: x:A ⟶ B[x],
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,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cat-ob_wf,
cat-arrow_wf,
nat-trans_wf,
cat-functor_wf,
small-category_wf,
functor-ob_wf,
cat-comp_wf,
functor-arrow_wf,
equal_wf,
squash_wf,
true_wf,
functor-arrow-comp,
iff_weakening_equal,
nat-trans-equation
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
applyEquality,
functionEquality,
setElimination,
rename,
natural_numberEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C,D,E:SmallCategory]. \mforall{}[F,G:Functor(C;D)]. \mforall{}[J:Functor(D;E)]. \mforall{}[tFG:nat-trans(C;D;F;G)].
\mforall{}[A,B:cat-ob(C)]. \mforall{}[g:cat-arrow(C) A B]. \mforall{}[v:cat-ob(E)].
((cat-comp(E) (ob(J) (ob(F) A)) (ob(J) (ob(G) B)) v
(cat-comp(E) (ob(J) (ob(F) A)) (ob(J) (ob(G) A)) (ob(J) (ob(G) B))
(arrow(J) (ob(F) A) (ob(G) A) (tFG A))
(arrow(J) (ob(G) A) (ob(G) B) (arrow(G) A B g))))
= (cat-comp(E) (ob(J) (ob(F) A)) (ob(J) (ob(G) B)) v
(cat-comp(E) (ob(J) (ob(F) A)) (ob(J) (ob(F) B)) (ob(J) (ob(G) B))
(arrow(J) (ob(F) A) (ob(F) B) (arrow(F) A B g))
(arrow(J) (ob(F) B) (ob(G) B) (tFG B)))))
Date html generated:
2017_10_05-AM-00_46_05
Last ObjectModification:
2017_07_28-AM-09_19_15
Theory : small!categories
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