Nuprl Lemma : presheaf-type-ap-morph-comp-eq
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[I,J,K:cat-ob(C)]. ∀[f:cat-arrow(C) J I].
∀[g:cat-arrow(C) K J]. ∀[a:X(I)]. ∀[b:X(J)]. ∀[u:A(a)].
((u a f) b g) = (u a cat-comp(C) K J I g f) ∈ A(cat-comp(C) K J I g f(a)) supposing b = f(a) ∈ X(J)
Proof
Definitions occuring in Statement :
presheaf-type-ap-morph: (u a f),
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
true: True,
all: ∀x:A. B[x]
Lemmas referenced :
psc-restriction_wf,
ps_context_cumulativity2,
small-category-cumulativity-2,
presheaf-type-at_wf,
I_set_wf,
cat-arrow_wf,
presheaf-type-ap-morph-comp,
equal_wf,
cat-comp_wf,
presheaf-type-ap-morph_wf,
subtype_rel-equal,
psc-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
productIsType,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
productElimination,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
dependent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I,J,K:cat-ob(C)].
\mforall{}[f:cat-arrow(C) J I]. \mforall{}[g:cat-arrow(C) K J]. \mforall{}[a:X(I)]. \mforall{}[b:X(J)]. \mforall{}[u:A(a)].
((u a f) b g) = (u a cat-comp(C) K J I g f) supposing b = f(a)
Date html generated:
2020_05_20-PM-01_26_01
Last ObjectModification:
2020_04_01-PM-00_01_23
Theory : presheaf!models!of!type!theory
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