Step
*
1
1
2
2
of Lemma
A-open-box-image_wf
1. X : CubicalSet
2. A : {X ⊢ _}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) ⊆r nameset(K)
9. x : nameset(I)
10. ∀x:nameset([x / J]). (f x ∈ nameset(K))
11. i : ℕ2
12. nameset([x / J]) ⊆r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. (¬(x ∈ J))
∧ l_subset(Cname;J;I)
∧ ((∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
∧ (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
∧ (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈bx.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
16. ∀x:nameset(J). (f x ∈ nameset(K))
17. map(f;J) ∈ nameset(K) List
18. f x ∈ nameset(K)
19. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
20. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
⊢ (¬(f x ∈ map(f;J)))
∧ l_subset(Cname;map(f;J);K)
∧ ((∀y:nameset(map(f;J)). ∀c:ℕ2.
(∃f∈A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f) = <y, c> ∈ (nameset(K) × ℕ2)))
∧ (∃f@0∈A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f@0) = <f x, i> ∈ (nameset(K) × ℕ2))
∧ (∀f@0∈A-open-box-image(X;A;I;K;f;alpha;bx).¬(A-face-name(f@0) = <f x, 1 - i> ∈ (nameset(K) × ℕ2))))
∧ (∀f@0∈A-open-box-image(X;A;I;K;f;alpha;bx).(fst(f@0) ∈ [f x / map(f;J)]))
∧ (∀f1,f2∈A-open-box-image(X;A;I;K;f;alpha;bx). ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(K) × ℕ2)))
BY
{ RepeatFor 2 (ParallelOp -6) }
1
1. X : CubicalSet
2. A : {X ⊢ _}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) ⊆r nameset(K)
9. x : nameset(I)
10. ∀x:nameset([x / J]). (f x ∈ nameset(K))
11. i : ℕ2
12. nameset([x / J]) ⊆r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. l_subset(Cname;J;I)
∧ ((∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
∧ (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
∧ (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈bx.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
16. ∀x:nameset(J). (f x ∈ nameset(K))
17. map(f;J) ∈ nameset(K) List
18. f x ∈ nameset(K)
19. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
20. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
21. (f x ∈ map(f;J))
⊢ (x ∈ J)
2
1. X : CubicalSet
2. A : {X ⊢ _}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) ⊆r nameset(K)
9. x : nameset(I)
10. ∀x:nameset([x / J]). (f x ∈ nameset(K))
11. i : ℕ2
12. nameset([x / J]) ⊆r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. ¬(x ∈ J)
16. ((∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
∧ (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
∧ (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈bx.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
17. l_subset(Cname;J;I)
18. ∀x:nameset(J). (f x ∈ nameset(K))
19. map(f;J) ∈ nameset(K) List
20. f x ∈ nameset(K)
21. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
22. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
⊢ l_subset(Cname;map(f;J);K)
3
1. X : CubicalSet
2. A : {X ⊢ _}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) ⊆r nameset(K)
9. x : nameset(I)
10. ∀x:nameset([x / J]). (f x ∈ nameset(K))
11. i : ℕ2
12. nameset([x / J]) ⊆r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. ¬(x ∈ J)
16. l_subset(Cname;J;I)
17. ((∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2)))
∧ (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
∧ (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2))))
∧ (∀f∈bx.(fst(f) ∈ [x / J]))
∧ (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
18. ∀x:nameset(J). (f x ∈ nameset(K))
19. map(f;J) ∈ nameset(K) List
20. f x ∈ nameset(K)
21. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
22. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
⊢ ((∀y:nameset(map(f;J)). ∀c:ℕ2.
(∃f∈A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f) = <y, c> ∈ (nameset(K) × ℕ2)))
∧ (∃f@0∈A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f@0) = <f x, i> ∈ (nameset(K) × ℕ2))
∧ (∀f@0∈A-open-box-image(X;A;I;K;f;alpha;bx).¬(A-face-name(f@0) = <f x, 1 - i> ∈ (nameset(K) × ℕ2))))
∧ (∀f@0∈A-open-box-image(X;A;I;K;f;alpha;bx).(fst(f@0) ∈ [f x / map(f;J)]))
∧ (∀f1,f2∈A-open-box-image(X;A;I;K;f;alpha;bx). ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(K) × ℕ2)))
Latex:
Latex:
1. X : CubicalSet
2. A : \{X \mvdash{} \_\}
3. I : Cname List
4. J : Cname List
5. K : Cname List
6. alpha : X(I)
7. f : name-morph(I;K)
8. nameset(map(f;J)) \msubseteq{}r nameset(K)
9. x : nameset(I)
10. \mforall{}x:nameset([x / J]). (f x \mmember{} nameset(K))
11. i : \mBbbN{}2
12. nameset([x / J]) \msubseteq{}r name-morph-domain(f;I)
13. bx : A-face(X;A;I;alpha) List
14. A-adjacent-compatible(X;A;I;alpha;bx)
15. (\mneg{}(x \mmember{} J))
\mwedge{} l\_subset(Cname;J;I)
\mwedge{} ((\mforall{}y:nameset(J). \mforall{}c:\mBbbN{}2. (\mexists{}f\mmember{}bx. A-face-name(f) = <y, c>))
\mwedge{} (\mexists{}f\mmember{}bx. A-face-name(f) = <x, i>)
\mwedge{} (\mforall{}f\mmember{}bx.\mneg{}(A-face-name(f) = <x, 1 - i>)))
\mwedge{} (\mforall{}f\mmember{}bx.(fst(f) \mmember{} [x / J]))
\mwedge{} (\mforall{}f1,f2\mmember{}bx. \mneg{}(A-face-name(f1) = A-face-name(f2)))
16. \mforall{}x:nameset(J). (f x \mmember{} nameset(K))
17. map(f;J) \mmember{} nameset(K) List
18. f x \mmember{} nameset(K)
19. \mforall{}fc:A-face(X;A;I;alpha). ((fc \mmember{} bx) {}\mRightarrow{} (fst(fc) \mmember{} [x / J]))
20. \mforall{}fc:A-face(X;A;I;alpha). ((fc \mmember{} bx) {}\mRightarrow{} (\muparrow{}isname(f (fst(fc)))))
\mvdash{} (\mneg{}(f x \mmember{} map(f;J)))
\mwedge{} l\_subset(Cname;map(f;J);K)
\mwedge{} ((\mforall{}y:nameset(map(f;J)). \mforall{}c:\mBbbN{}2.
(\mexists{}f\mmember{}A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f) = <y, c>))
\mwedge{} (\mexists{}f@0\mmember{}A-open-box-image(X;A;I;K;f;alpha;bx). A-face-name(f@0) = <f x, i>)
\mwedge{} (\mforall{}f@0\mmember{}A-open-box-image(X;A;I;K;f;alpha;bx).\mneg{}(A-face-name(f@0) = <f x, 1 - i>)))
\mwedge{} (\mforall{}f@0\mmember{}A-open-box-image(X;A;I;K;f;alpha;bx).(fst(f@0) \mmember{} [f x / map(f;J)]))
\mwedge{} (\mforall{}f1,f2\mmember{}A-open-box-image(X;A;I;K;f;alpha;bx). \mneg{}(A-face-name(f1) = A-face-name(f2)))
By
Latex:
RepeatFor 2 (ParallelOp -6)
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