Nuprl Lemma : presheaf-sigma-intro
∀C:SmallCategory. ∀G:ps_context{j:l}(C). ∀A:{G ⊢ _}. ∀B:{G.A ⊢ _}. ((∃a:{G ⊢ _:A}. {G ⊢ _:(B)[a]}) ⇒ {G ⊢ _:Σ A B})
Proof
Definitions occuring in Statement :
presheaf-sigma: Σ A B,
pscm-id-adjoin: [u],
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
ps_context: __⊢,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
presheaf-pair_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
psc-adjoin_wf,
presheaf-term_wf,
pscm-ap-type_wf,
pscm-id-adjoin_wf,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
rename,
introduction,
cut,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
productIsType,
universeIsType
Latex:
\mforall{}C:SmallCategory. \mforall{}G:ps\_context\{j:l\}(C). \mforall{}A:\{G \mvdash{} \_\}. \mforall{}B:\{G.A \mvdash{} \_\}.
((\mexists{}a:\{G \mvdash{} \_:A\}. \{G \mvdash{} \_:(B)[a]\}) {}\mRightarrow{} \{G \mvdash{} \_:\mSigma{} A B\})
Date html generated:
2020_05_20-PM-01_35_35
Last ObjectModification:
2020_04_02-PM-06_35_37
Theory : presheaf!models!of!type!theory
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