Nuprl Lemma : presheaf-sigma-p-p
∀C:SmallCategory. ∀X:ps_context{j:l}(C). ∀T,A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀E:{X.T ⊢ _}.
(((Σ A B)p)p = Σ ((A)p)p (B)(p o p o p;q) ∈ {X.T.E ⊢ _})
Proof
Definitions occuring in Statement :
presheaf-sigma: Σ A B,
pscm-adjoin: (s;u),
psc-snd: q,
psc-fst: p,
psc-adjoin: X.A,
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
pscm-comp: G o F,
ps_context: __⊢,
all: ∀x:A. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
presheaf-type: {X ⊢ _},
psc-snd: q,
psc-fst: p,
pscm-comp: G o F,
pscm-ap-type: (AF)s,
pscm-adjoin: (s;u),
presheaf-sigma: Σ A B,
compose: f o g,
pscm-ap: (s)x,
psc-adjoin-set: (v;u)
Lemmas referenced :
pscm-presheaf-sigma,
ps_context_cumulativity2,
psc-adjoin_wf,
presheaf-type-cumulativity2,
pscm-comp_wf,
psc-fst_wf,
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
presheaf-type_wf,
small-category-cumulativity-2,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
isectElimination,
because_Cache,
hypothesis,
sqequalRule,
setElimination,
rename,
productElimination,
Error :memTop,
universeIsType,
inhabitedIsType
Latex:
\mforall{}C:SmallCategory. \mforall{}X:ps\_context\{j:l\}(C). \mforall{}T,A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}E:\{X.T \mvdash{} \_\}.
(((\mSigma{} A B)p)p = \mSigma{} ((A)p)p (B)(p o p o p;q))
Date html generated:
2020_05_20-PM-01_31_34
Last ObjectModification:
2020_04_02-PM-05_49_42
Theory : presheaf!models!of!type!theory
Home
Index