Nuprl Lemma : equiv_term-0
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}].
∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[a:{G ⊢ _:A}]. ∀[t,c:Top]. ∀[cA:G +⊢ Compositon(A)].
∀[cT:G +⊢ Compositon(T)].
(equiv f [phi ⊢→ (t,c)] a
= transprt(G;(fiber-comp(G;T;A;equiv-fun(f);a;cT;cA))p;contr-center(equiv-contr(f;a)))
∈ {G ⊢ _:Fiber(equiv-fun(f);a)})
supposing phi = 0(𝔽) ∈ {G ⊢ _:𝔽}
Proof
Definitions occuring in Statement :
equiv_term: equiv f [phi ⊢→ (t,c)] a,
fiber-comp: fiber-comp(X;T;A;w;a;cT;cA),
transprt: transprt(G;cA;a0),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
equiv-contr: equiv-contr(f;a),
equiv-fun: equiv-fun(f),
cubical-equiv: Equiv(T;A),
cubical-fiber: Fiber(w;a),
contr-center: contr-center(c),
face-0: 0(𝔽),
face-type: 𝔽,
interval-type: 𝕀,
cc-fst: p,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
equiv_term: equiv f [phi ⊢→ (t,c)] a,
subtype_rel: A ⊆r B
Lemmas referenced :
equiv-term-0,
fiber-comp_wf,
equiv-fun_wf,
composition-structure_wf,
cubical_set_cumulativity-i-j,
istype-top,
istype-cubical-term,
cubical-equiv_wf,
cubical-type_wf,
face-0_wf,
face-type_wf,
cubical_set_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
universeIsType,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
equalityIstype
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}].
\mforall{}[A,T:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(T;A)\}]. \mforall{}[a:\{G \mvdash{} \_:A\}]. \mforall{}[t,c:Top]. \mforall{}[cA:G +\mvdash{} Compositon(A)].
\mforall{}[cT:G +\mvdash{} Compositon(T)].
(equiv f [phi \mvdash{}\mrightarrow{} (t,c)] a
= transprt(G;(fiber-comp(G;T;A;equiv-fun(f);a;cT;cA))p;contr-center(equiv-contr(f;a))))
supposing phi = 0(\mBbbF{})
Date html generated:
2020_05_20-PM-05_36_33
Last ObjectModification:
2020_04_21-AM-09_44_20
Theory : cubical!type!theory
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