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


1. fset(ℕ)
2. phi : 𝔽(I)
3. {I,phi ⊢ _:𝔽}
4. fset(ℕ)
5. J ⟶ I
6. (phi g) 1
7. 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. fs ∈ fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} z ∈ v} )
⊢ (\/(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
((Assert g ∈ I,phi(J) BY
          (RepUR ``cubical-subset rep-sub-sheaf names-cat cat-arrow`` THEN Auto))
   THEN Assert ⌜∀p1,p2:fset(names(I)).
                  ((↑fset-disjoint(NamesDeq;p1;p2))  <p1, p2> ∈  (irr-face-morph(I;p1;p2) ∈ I,phi(I)))⌝⋅
   }

1
.....assertion..... 
1. fset(ℕ)
2. phi : 𝔽(I)
3. {I,phi ⊢ _:𝔽}
4. fset(ℕ)
5. J ⟶ I
6. (phi g) 1
7. 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. fs ∈ fset({z:{p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} z ∈ v} )
11. g ∈ I,phi(J)
⊢ ∀p1,p2:fset(names(I)).  ((↑fset-disjoint(NamesDeq;p1;p2))  <p1, p2> ∈  (irr-face-morph(I;p1;p2) ∈ I,phi(I)))

2
1. fset(ℕ)
2. phi : 𝔽(I)
3. {I,phi ⊢ _:𝔽}
4. fset(ℕ)
5. J ⟶ I
6. (phi g) 1
7. 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. 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> ∈  (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))


Latex:


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
\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


Latex:
((Assert  g  \mmember{}  I,phi(J)  BY
                (RepUR  ``cubical-subset  rep-sub-sheaf  names-cat  cat-arrow``  0  THEN  Auto))
  THEN  Assert  \mkleeneopen{}\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)))\mkleeneclose{}\mcdot{}
  )




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