Nuprl Lemma : fl-morph-comp-1
∀[J,K:fset(ℕ)]. ∀[f:K ⟶ J]. ∀[z:Point(dM(J))]. ∀[h:dma-hom(dM(J);dM(K))].
(dM-to-FL(J;z))<f> = dM-to-FL(K;h z) ∈ Point(face_lattice(K)) supposing ∀i:names(J). ((h <i>) = (f i) ∈ Point(dM(K)))
Proof
Definitions occuring in Statement :
fl-morph: <f>,
dM-to-FL: dM-to-FL(I;z),
face_lattice: face_lattice(I),
names-hom: I ⟶ J,
dM_inc: <x>,
dM: dM(I),
names: names(I),
dma-hom: dma-hom(dma1;dma2),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
guard: {T},
so_apply: x[s],
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
names-hom: I ⟶ J,
all: ∀x:A. B[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
dM: dM(I),
compose: f o g,
cand: A c∧ B,
fl-morph: <f>,
lattice-point: Point(l),
record-select: r.x,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
fl1: (x=1),
fl0: (x=0),
dm-neg: ¬(x),
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,
dma-neg: ¬(x)
Lemmas referenced :
all_wf,
names_wf,
equal_wf,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM_inc_wf,
dma-hom_wf,
names-hom_wf,
fset_wf,
nat_wf,
dM-hom-unique,
face_lattice_wf,
face_lattice-deq_wf,
compose-bounded-lattice-hom,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
fl-morph_wf,
dM-to-FL-is-hom,
subtype_rel-equal,
bounded-lattice-hom_wf,
free-DeMorgan-lattice_wf,
names-deq_wf,
squash_wf,
true_wf,
free-dma-hom-is-lattice-hom,
iff_weakening_equal,
dM-to-FL-inc,
dM-to-FL_wf,
fl-lift_wf,
dm-neg_wf,
dM-to-FL-neg2,
lattice-0_wf,
set_wf,
face-lattice_wf,
face-lattice0_wf,
face-lattice1_wf,
dM-to-FL-opp,
dM_opp_wf,
neg-dM_inc,
dma-neg-dM_inc,
dma-neg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
instantiate,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
because_Cache,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
lambdaFormation,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[J,K:fset(\mBbbN{})]. \mforall{}[f:K {}\mrightarrow{} J]. \mforall{}[z:Point(dM(J))]. \mforall{}[h:dma-hom(dM(J);dM(K))].
(dM-to-FL(J;z))<f> = dM-to-FL(K;h z) supposing \mforall{}i:names(J). ((h <i>) = (f i))
Date html generated:
2017_10_05-AM-01_14_10
Last ObjectModification:
2017_07_28-AM-09_31_23
Theory : cubical!type!theory
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