Nuprl Lemma : ps-sigma-elim-unelim
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}].
(SigmaUnElim o SigmaElim = 1(X.Σ A B) ∈ psc_map{[i | j]:l}(C; X.Σ A B; X.Σ A B))
Proof
Definitions occuring in Statement :
sigma-unelim-pscm: SigmaUnElim,
sigma-elim-pscm: SigmaElim,
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
presheaf-type: {X ⊢ _},
pscm-id: 1(X),
pscm-comp: G o F,
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
psc-adjoin: X.A,
all: ∀x:A. B[x],
presheaf-sigma: Σ A B,
pscm-id: 1(X),
sigma-unelim-pscm: SigmaUnElim,
sigma-elim-pscm: SigmaElim,
pscm-comp: G o F,
compose: f o g,
spreadn: spread3,
psc-adjoin-set: (v;u),
I_set: A(I),
functor-ob: ob(F),
pi1: fst(t)
Lemmas referenced :
presheaf-type_wf,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
ps_context_wf,
small-category_wf,
presheaf-sigma_wf,
pscm-comp_wf,
sigma-elim-pscm_wf,
sigma-unelim-pscm_wf,
pscm-id_wf,
I_set_wf,
cat-ob_wf,
pscm-equal,
I_set_pair_redex_lemma,
presheaf_type_at_pair_lemma,
presheaf-type-at_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
functionExtensionality,
independent_isectElimination,
dependent_functionElimination,
Error :memTop,
productElimination,
dependent_pairEquality_alt,
productIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}].
(SigmaUnElim o SigmaElim = 1(X.\mSigma{} A B))
Date html generated:
2020_05_20-PM-01_32_41
Last ObjectModification:
2020_04_02-PM-06_48_09
Theory : presheaf!models!of!type!theory
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