Nuprl Lemma : pscm-presheaf-sigma
∀C:SmallCategory. ∀X,Delta:ps_context{j:l}(C). ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:psc_map{j:l}(C; Delta; X).
((Σ A B)s = Σ (A)s (B)(s o p;q) ∈ {Delta ⊢ _})
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,
psc_map: A ⟶ B,
ps_context: __⊢,
all: ∀x:A. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
cat-comp: cat-comp(C),
compose: f o g,
presheaf-type: {X ⊢ _},
psc-snd: q,
pscm-ap-type: (AF)s,
psc-fst: p,
pscm-comp: G o F,
pscm-ap: (s)x,
uimplies: b supposing a,
presheaf-sigma: Σ A B,
and: P ∧ Q,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
presheaf-type-at: A(a)
Lemmas referenced :
presheaf-type-equal,
pscm-ap-type_wf,
presheaf-sigma_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-adjoin_wf,
pscm-comp_wf,
psc-fst_wf,
subtype_rel_self,
psc_map_wf,
psc-snd_wf,
small-category-cumulativity-2,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
presheaf_type_at_pair_lemma,
presheaf_type_ap_morph_pair_lemma,
pscm-ap_wf,
pscm-adjoin-ap,
csm_comp_fst_adjoin_set_lemma,
cc_snd_adjoin_set_lemma,
psc-adjoin-set_wf,
I_set_wf,
cat-ob_wf,
subtype_rel-equal,
psc-restriction_wf,
equal_wf,
pscm-ap-restriction,
iff_weakening_equal,
pscm-ap-pscm-comp,
cc_fst_adjoin_set_lemma,
psc-adjoin-set-restriction,
cat-arrow_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
instantiate,
applyEquality,
sqequalRule,
setElimination,
rename,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
inhabitedIsType,
dependent_functionElimination,
Error :memTop,
dependent_pairEquality_alt,
lambdaEquality_alt,
productEquality,
functionExtensionality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
productIsType,
functionIsType
Latex:
\mforall{}C:SmallCategory. \mforall{}X,Delta:ps\_context\{j:l\}(C). \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}.
\mforall{}s:psc\_map\{j:l\}(C; Delta; X).
((\mSigma{} A B)s = \mSigma{} (A)s (B)(s o p;q))
Date html generated:
2020_05_20-PM-01_31_27
Last ObjectModification:
2020_04_02-PM-05_54_00
Theory : presheaf!models!of!type!theory
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