Nuprl Lemma : trans-horizontal-comp_wf
∀[C,D,E:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[J,K:Functor(D;E)]. ∀[tFG:nat-trans(C;D;F;G)]. ∀[tJK:nat-trans(D;E;J;K)].
(trans-horizontal-comp(E;F;G;J;K;tFG;tJK) ∈ nat-trans(C;E;functor-comp(F;J);functor-comp(G;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),
nat-trans: nat-trans(C;D;F;G),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
trans-horizontal-comp: trans-horizontal-comp(E;F;G;J;K;tFG;tJK),
so_lambda: λ2x.t[x],
nat-trans: nat-trans(C;D;F;G),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
functor-ob: ob(F),
pi1: fst(t),
functor-comp: functor-comp(F;G),
mk-functor: mk-functor,
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
mk-nat-trans_wf,
functor-comp_wf,
cat-comp_wf,
functor-ob_wf,
functor-arrow_wf,
subtype_rel-equal,
cat-arrow_wf,
cat-ob_wf,
ob_mk_functor_lemma,
arrow_mk_functor_lemma,
nat-trans_wf,
cat-functor_wf,
small-category_wf,
equal_wf,
squash_wf,
true_wf,
nat-trans-equation,
nat-trans-assoc-equation,
cat-comp-assoc,
nat-trans-comp-equation2,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
lambdaEquality,
applyEquality,
setElimination,
rename,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[C,D,E:SmallCategory]. \mforall{}[F,G:Functor(C;D)]. \mforall{}[J,K:Functor(D;E)]. \mforall{}[tFG:nat-trans(C;D;F;G)].
\mforall{}[tJK:nat-trans(D;E;J;K)].
(trans-horizontal-comp(E;F;G;J;K;tFG;tJK) \mmember{} nat-trans(C;E;functor-comp(F;J);functor-comp(G;K)))
Date html generated:
2017_10_05-AM-00_48_00
Last ObjectModification:
2017_07_28-AM-09_19_54
Theory : small!categories
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