Nuprl Lemma : fiber-member-transprt-const-fiber-comp
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[pr:{X ⊢ _:Fiber(w;a)}]. ∀[cT:X +⊢ Compositon(T)].
∀[cA:X +⊢ Compositon(A)].
(fiber-member(transprt-const(X;fiber-comp(X;T;A;w;a;cT;cA);pr)) = transprt-const(X;cT;pr.1) ∈ {X ⊢ _:T})
Proof
Definitions occuring in Statement :
fiber-comp: fiber-comp(X;T;A;w;a;cT;cA),
transprt-const: transprt-const(G;cA;a),
composition-structure: Gamma ⊢ Compositon(A),
fiber-member: fiber-member(p),
cubical-fiber: Fiber(w;a),
cubical-fst: p.1,
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
fiber-comp: fiber-comp(X;T;A;w;a;cT;cA),
fiber-member: fiber-member(p),
member: t ∈ T,
subtype_rel: A ⊆r B,
cubical-fiber: Fiber(w;a),
all: ∀x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a
Lemmas referenced :
fst-transprt-const-sigma,
path-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
path_comp_wf,
csm-comp-structure_wf,
composition-structure_wf,
istype-cubical-term,
cubical-fiber_wf,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf,
cc-snd_wf,
csm-cubical-fun,
cubical-term-eqcd,
cubical-app_wf_fun
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeIsType,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[pr:\{X \mvdash{} \_:Fiber(w;a)\}].
\mforall{}[cT:X +\mvdash{} Compositon(T)]. \mforall{}[cA:X +\mvdash{} Compositon(A)].
(fiber-member(transprt-const(X;fiber-comp(X;T;A;w;a;cT;cA);pr)) = transprt-const(X;cT;pr.1))
Date html generated:
2020_05_20-PM-05_13_49
Last ObjectModification:
2020_04_18-PM-00_33_23
Theory : cubical!type!theory
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