Nuprl Lemma : nat-trans-equal2
∀[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[A,B:nat-trans(C;D;F;G)].
A = B ∈ nat-trans(C;D;F;G) supposing A = B ∈ (A:cat-ob(C) ⟶ (cat-arrow(D) (F A) (G A)))
Proof
Definitions occuring in Statement :
nat-trans: nat-trans(C;D;F;G),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
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,
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat-trans: nat-trans(C;D;F;G),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
cat-arrow_wf,
functor-ob_wf,
nat-trans-equation,
cat-comp_wf,
functor-arrow_wf,
cat-ob_wf,
iff_weakening_equal,
all_wf,
nat-trans_wf,
cat-functor_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
functionExtensionality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
functionEquality,
setElimination,
rename,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G:Functor(C;D)]. \mforall{}[A,B:nat-trans(C;D;F;G)]. A = B supposing A = B
Date html generated:
2020_05_20-AM-07_51_26
Last ObjectModification:
2017_07_28-AM-09_19_17
Theory : small!categories
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