Nuprl Lemma : trans-exchange
∀[C,D,E:SmallCategory]. ∀[F,G,H:Functor(C;D)]. ∀[I,J,K:Functor(D;E)]. ∀[tFG:nat-trans(C;D;F;G)].
∀[tGH:nat-trans(C;D;G;H)]. ∀[tIJ:nat-trans(D;E;I;J)]. ∀[tJK:nat-trans(D;E;J;K)].
(trans-horizontal-comp(E;F;G;I;J;tFG;tIJ) o trans-horizontal-comp(E;G;H;J;K;tGH;tJK)
= trans-horizontal-comp(E;F;H;I;K;tFG o tGH;tIJ o tJK)
∈ nat-trans(C;E;functor-comp(F;I);functor-comp(H;K)))
Proof
Definitions occuring in Statement :
trans-horizontal-comp: trans-horizontal-comp(E;F;G;J;K;tFG;tJK),
functor-comp: functor-comp(F;G),
trans-comp: t1 o t2,
nat-trans: nat-trans(C;D;F;G),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
trans-comp: t1 o t2,
trans-horizontal-comp: trans-horizontal-comp(E;F;G;J;K;tFG;tJK),
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
functor-comp: functor-comp(F;G),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
nat-trans: nat-trans(C;D;F;G),
true: True,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
nat-trans-equal2,
functor-comp_wf,
trans-comp_wf,
trans-horizontal-comp_wf,
cat-ob_wf,
nat-trans_wf,
cat-functor_wf,
small-category_wf,
ap_mk_nat_trans_lemma,
ob_mk_functor_lemma,
cat-arrow_wf,
functor-ob_wf,
cat-comp_wf,
functor-arrow_wf,
equal_wf,
squash_wf,
true_wf,
nat-trans-equation,
cat-comp-assoc,
functor-arrow-comp,
nat-trans-assoc-equation,
nat-trans-comp-equation,
nat-trans-assoc-comp-equation,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
functionExtensionality,
because_Cache,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
setElimination,
rename,
natural_numberEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C,D,E:SmallCategory]. \mforall{}[F,G,H:Functor(C;D)]. \mforall{}[I,J,K:Functor(D;E)]. \mforall{}[tFG:nat-trans(C;D;F;G)].
\mforall{}[tGH:nat-trans(C;D;G;H)]. \mforall{}[tIJ:nat-trans(D;E;I;J)]. \mforall{}[tJK:nat-trans(D;E;J;K)].
(trans-horizontal-comp(E;F;G;I;J;tFG;tIJ) o trans-horizontal-comp(E;G;H;J;K;tGH;tJK)
= trans-horizontal-comp(E;F;H;I;K;tFG o tGH;tIJ o tJK))
Date html generated:
2017_10_05-AM-00_48_04
Last ObjectModification:
2017_07_28-AM-09_19_56
Theory : small!categories
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