Nuprl Lemma : nat-trans-equal
∀[C,D:SmallCategory]. ∀[F,G:Functor(C;D)]. ∀[A:nat-trans(C;D;F;G)]. ∀[B:A:cat-ob(C) ⟶ (cat-arrow(D) (functor-ob(F) A)
(functor-ob(G) A))].
A = B ∈ nat-trans(C;D;F;G) supposing A = B ∈ (A:cat-ob(C) ⟶ (cat-arrow(D) (functor-ob(F) A) (functor-ob(G) A)))
Proof
Definitions occuring in Statement :
nat-trans: nat-trans(C;D;F;G),
functor-ob: functor-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 :
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
nat-trans: nat-trans(C;D;F;G),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nat-trans_wf,
functor-arrow_wf,
cat-comp_wf,
functor-ob_wf,
equal_wf,
cat-arrow_wf,
cat-ob_wf,
all_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
isect_memberEquality,
functionEquality,
because_Cache,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
lemma_by_obid,
hypothesis,
dependent_set_memberEquality,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[C,D:SmallCategory]. \mforall{}[F,G:Functor(C;D)]. \mforall{}[A:nat-trans(C;D;F;G)].
\mforall{}[B:A:cat-ob(C) {}\mrightarrow{} (cat-arrow(D) (functor-ob(F) A) (functor-ob(G) A))].
A = B supposing A = B
Date html generated:
2016_05_18-AM-11_52_32
Last ObjectModification:
2015_12_28-PM-02_25_09
Theory : small!categories
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