Nuprl Lemma : sub_ps_context_transitivity
∀[C:SmallCategory]. ∀[X,Y,Z:ps_context{j:l}(C)].
sub_ps_context{j:l}(C; Z; X) supposing sub_ps_context{j:l}(C; Z; Y) ∧ sub_ps_context{j:l}(C; Y; X)
Proof
Definitions occuring in Statement :
sub_ps_context: Y ⊆ X,
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
sub_ps_context: Y ⊆ X,
subtype_rel: A ⊆r B,
guard: {T},
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,
all: ∀x:A. B[x],
cat-comp: cat-comp(C),
compose: f o g,
pscm-id: 1(X),
pscm-comp: G o F
Lemmas referenced :
pscm-comp_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
subtype_rel_self,
psc_map_wf,
sub_ps_context_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
productIsType,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X,Y,Z:ps\_context\{j:l\}(C)].
sub\_ps\_context\{j:l\}(C; Z; X) supposing sub\_ps\_context\{j:l\}(C; Z; Y) \mwedge{} sub\_ps\_context\{j:l\}(C; Y; X)
Date html generated:
2020_05_20-PM-01_24_43
Last ObjectModification:
2020_04_01-AM-11_00_41
Theory : presheaf!models!of!type!theory
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