Nuprl Lemma : fl-morph-comp
∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J].
(<f ⋅ g> = (<g> o <f>) ∈ (Point(face_lattice(I)) ⟶ Point(face_lattice(K))))
Proof
Definitions occuring in Statement :
fl-morph: <f>,
face_lattice: face_lattice(I),
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
lattice-point: Point(l),
fset: fset(T),
compose: f o g,
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
face_lattice: face_lattice(I),
subtype_rel: A ⊆r B,
names-hom: I ⟶ J,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
all: ∀x:A. B[x],
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
compose: f o g,
fl-morph: <f>,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
fl1: (x=1),
fl0: (x=0),
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
top: Top,
dma-hom: dma-hom(dma1;dma2),
dM_inc: <x>
Lemmas referenced :
fl-lift-unique,
names_wf,
names-deq_wf,
face-lattice_wf,
face_lattice-deq_wf,
dM-to-FL_wf,
dm-neg_wf,
nh-comp_wf,
names-hom_wf,
subtype_rel-equal,
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
free-DeMorgan-lattice_wf,
squash_wf,
true_wf,
face_lattice_wf,
dM-to-FL-neg2,
lattice-0_wf,
bdd-distributive-lattice_wf,
iff_weakening_equal,
fset_wf,
nat_wf,
compose-bounded-lattice-hom,
bdd-distributive-lattice-subtype-bdd-lattice,
fl-morph_wf,
bounded-lattice-hom_wf,
equal_functionality_wrt_subtype_rel2,
fl-morph-fl0,
free-dma-lift_wf,
dM-point,
free-dl-point,
free-dml-deq_wf,
set_wf,
dma-hom_wf,
free-DeMorgan-algebra_wf,
all_wf,
free-dma-point,
dminc_wf,
free-dma-neg,
fl-morph-comp-1,
fl-morph-fl1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
because_Cache,
applyEquality,
sqequalRule,
instantiate,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
functionEquality,
independent_pairFormation,
voidElimination,
voidEquality,
hyp_replacement,
applyLambdaEquality,
promote_hyp
Latex:
\mforall{}[I,J,K:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[g:K {}\mrightarrow{} J]. (<f \mcdot{} g> = (<g> o <f>))
Date html generated:
2017_10_05-AM-01_14_23
Last ObjectModification:
2017_07_28-AM-09_31_32
Theory : cubical!type!theory
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