Nuprl Lemma : fl-morph-restriction
∀[I,J,K:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[g:J ⟶ K]. ∀[phi:𝔽(K)]. ((g(phi))<f> = g ⋅ f(phi) ∈ 𝔽(I))
Proof
Definitions occuring in Statement :
face-presheaf: 𝔽,
fl-morph: <f>,
cube-set-restriction: f(s),
I_cube: A(I),
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
squash: ↓T,
fl-morph: <f>,
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
cube-set-restriction: f(s),
pi2: snd(t),
face-presheaf: 𝔽,
true: True
Lemmas referenced :
cube-set-restriction-comp,
face-presheaf_wf2,
equal_wf,
squash_wf,
true_wf,
istype-universe,
I_cube_wf,
cube-set-restriction_wf,
names-hom_wf,
fset_wf,
nat_wf,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
hypothesisEquality,
hyp_replacement,
equalitySymmetry,
sqequalRule,
applyEquality,
instantiate,
lambdaEquality_alt,
imageElimination,
isectElimination,
equalityTransitivity,
universeIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[I,J,K:fset(\mBbbN{})]. \mforall{}[f:I {}\mrightarrow{} J]. \mforall{}[g:J {}\mrightarrow{} K]. \mforall{}[phi:\mBbbF{}(K)]. ((g(phi))<f> = g \mcdot{} f(phi))
Date html generated:
2020_05_20-PM-01_44_57
Last ObjectModification:
2020_04_19-PM-01_55_27
Theory : cubical!type!theory
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