Nuprl Lemma : presheaf-pi-implies-sigma
∀C:SmallCategory. ∀G:ps_context{j:l}(C). ∀A:{G ⊢ _}. ∀B:{G.A ⊢ _}. ({G ⊢ _:A} ⇒ {G ⊢ _:ΠA B} ⇒ {G ⊢ _:Σ A B})
Proof
Definitions occuring in Statement :
presheaf-sigma: Σ A B,
presheaf-pi: ΠA B,
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
ps_context: __⊢,
all: ∀x:A. B[x],
implies: P ⇒ Q,
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
presheaf-sigma-intro,
presheaf-app_wf,
presheaf-term_wf,
pscm-ap-type_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-id-adjoin_wf,
presheaf-pi_wf,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
dependent_pairFormation_alt,
isectElimination,
because_Cache,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
applyEquality,
sqequalRule
Latex:
\mforall{}C:SmallCategory. \mforall{}G:ps\_context\{j:l\}(C). \mforall{}A:\{G \mvdash{} \_\}. \mforall{}B:\{G.A \mvdash{} \_\}.
(\{G \mvdash{} \_:A\} {}\mRightarrow{} \{G \mvdash{} \_:\mPi{}A B\} {}\mRightarrow{} \{G \mvdash{} \_:\mSigma{} A B\})
Date html generated:
2020_05_20-PM-01_35_39
Last ObjectModification:
2020_04_03-AM-01_19_09
Theory : presheaf!models!of!type!theory
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