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Step * 1 2 of Lemma extend-face-term-property


1. I : fset(ℕ)
2. phi : 𝔽(I)
3. u : {I,phi ⊢ _:𝔽}
4. J : fset(ℕ)
5. g : J ⟶ I
6. (phi g) = 1
7. v : fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
8. phi = \/(λpr.irr_face(I;fst(pr);snd(pr))"(v)) ∈ Point(face_lattice(I))
9. fs : fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
10. v = fs ∈ fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
11. g ∈ I,phi(J)
12. ∀p1,p2:fset(names(I)).  ((↑fset-disjoint(NamesDeq;p1;p2)) ⇒ <p1, p2> ∈ v ⇒ (irr-face-morph(I;p1;p2) ∈ I,phi(I)))
⊢ (\/(λpr.let as,bs = pr in irr_face(I;as;bs) ∧ u(irr-face-morph(I;as;bs))"(fs)))<g> = u(g) ∈ Point(face_lattice(J))
BY
{ ((InstLemma `lattice-hom-fset-join` [⌜face_lattice(I)⌝;⌜face_lattice(J)⌝;⌜face_lattice-deq()⌝;⌜face_lattice-deq()⌝]⋅
    THENA Auto
    )
   THEN (RWO "-1" 0 THENA Auto)
   ) }

1
1. I : fset(ℕ)
2. phi : 𝔽(I)
3. u : {I,phi ⊢ _:𝔽}
4. J : fset(ℕ)
5. g : J ⟶ I
6. (phi g) = 1
7. v : fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
8. phi = \/(λpr.irr_face(I;fst(pr);snd(pr))"(v)) ∈ Point(face_lattice(I))
9. fs : fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
10. v = fs ∈ fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} | z ∈ v} )
11. g ∈ I,phi(J)
12. ∀p1,p2:fset(names(I)).  ((↑fset-disjoint(NamesDeq;p1;p2)) ⇒ <p1, p2> ∈ v ⇒ (irr-face-morph(I;p1;p2) ∈ I,phi(I)))
13. ∀[f:Hom(face_lattice(I);face_lattice(J))]. ∀[s:fset(Point(face_lattice(I)))].
      ((f \/(s)) = \/(f"(s)) ∈ Point(face_lattice(J)))
⊢ \/(<g>"(λpr.let as,bs = pr in irr_face(I;as;bs) ∧ u(irr-face-morph(I;as;bs))"(fs))) = u(g) ∈ Point(face_lattice(J))

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Latex:

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Latex:

1.  I  :  fset(\mBbbN{})
2.  phi  :  \mBbbF{}(I)
3.  u  :  \{I,phi  \mvdash{}  \_:\mBbbF{}\}
4.  J  :  fset(\mBbbN{})
5.  g  :  J  {}\mrightarrow{}  I
6.  (phi  g)  =  1
7.  v  :  fset(\{p:fset(names(I))  \mtimes{}  fset(names(I))|  \muparrow{}fset-disjoint(NamesDeq;fst(p);snd(p))\}  )
8.  phi  =  \mbackslash{}/(\mlambda{}pr.irr\_face(I;fst(pr);snd(pr))"(v))
9.  fs  :  fset(\{z:\{p:fset(names(I))  \mtimes{}  fset(names(I))|  \muparrow{}fset-disjoint(NamesDeq;fst(p);snd(p))\}  |  z  \mmember{}  v\}\000C  )
10.  v  =  fs
11.  g  \mmember{}  I,phi(J)
12.  \mforall{}p1,p2:fset(names(I)).
            ((\muparrow{}fset-disjoint(NamesDeq;p1;p2))  {}\mRightarrow{}  <p1,  p2>  \mmember{}  v  {}\mRightarrow{}  (irr-face-morph(I;p1;p2)  \mmember{}  I,phi(I)))
\mvdash{}  (\mbackslash{}/(\mlambda{}pr.let  as,bs  =  pr  in  irr\_face(I;as;bs)  \mwedge{}  u(irr-face-morph(I;as;bs))"(fs)))<g>  =  u(g)


By

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Latex:
((InstLemma  `lattice-hom-fset-join`  [\mkleeneopen{}face\_lattice(I)\mkleeneclose{};\mkleeneopen{}face\_lattice(J)\mkleeneclose{};\mkleeneopen{}face\_lattice-deq()\mkleeneclose{};
    \mkleeneopen{}face\_lattice-deq()\mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  (RWO  "-1"  0  THENA  Auto)
  )

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