Nuprl Lemma : ps-sigma-elim-equality-rule2
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}].
∀[t1:{X.A.B ⊢ _:(T)SigmaUnElim}]. ∀[t2:{X.Σ A B ⊢ _:T}].
(t1)SigmaElim = t2 ∈ {X.Σ A B ⊢ _:T} supposing t1 = (t2)SigmaUnElim ∈ {X.A.B ⊢ _:(T)SigmaUnElim}
Proof
Definitions occuring in Statement :
sigma-unelim-pscm: SigmaUnElim,
sigma-elim-pscm: SigmaElim,
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
true: True,
presheaf-term-at: u(a)
Lemmas referenced :
pscm-ap-term_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
presheaf-sigma_wf,
sigma-unelim-pscm_wf,
presheaf-term_wf2,
pscm-ap-type_wf,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf,
ps-sigma-elim-rule,
equal_wf,
squash_wf,
true_wf,
istype-universe,
I_set_wf,
presheaf-term-equal,
ps-sigma-unelim-elim-term,
presheaf-term-at_wf,
cat-ob_wf,
presheaf-term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalityIstype,
because_Cache,
hypothesisEquality,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
hypothesis,
sqequalRule,
universeIsType,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
independent_isectElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[T:\{X.\mSigma{} A B \mvdash{} \_\}].
\mforall{}[t1:\{X.A.B \mvdash{} \_:(T)SigmaUnElim\}]. \mforall{}[t2:\{X.\mSigma{} A B \mvdash{} \_:T\}].
(t1)SigmaElim = t2 supposing t1 = (t2)SigmaUnElim
Date html generated:
2020_05_20-PM-01_33_09
Last ObjectModification:
2020_04_03-AM-11_03_16
Theory : presheaf!models!of!type!theory
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