Step
*
1
1
2
2
1
1
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. l_subset(Cname;J;I)
16. ∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
17. (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
18. (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
19. (∀f∈bx.(fst(f) ∈ [x / J]))
20. (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
21. ∀x:nameset(J). (f x ∈ nameset(K))
22. map(f;J) ∈ nameset(K) List
23. f x ∈ nameset(K)
24. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
25. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
26. y : nameset(J)
27. (y ∈ J)
28. (f x) = (f y) ∈ Cname
29. J ∈ nameset(J) List
30. f ∈ nameset(J) ⟶ Cname
⊢ (x ∈ J)
BY
{ (Assert x = y ∈ nameset(I) BY
(DVar `f' THEN BackThruSomeHyp)) }
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 : nameset(I) ⟶ extd-nameset(K)
8. ∀i,j:nameset(I). ((↑isname(f i)) ⇒ (↑isname(f j)) ⇒ ((f i) = (f j) ∈ extd-nameset(K)) ⇒ (i = j ∈ nameset(I)))
9. nameset(map(f;J)) ⊆r nameset(K)
10. x : nameset(I)
11. ∀x:nameset([x / J]). (f x ∈ nameset(K))
12. i : ℕ2
13. nameset([x / J]) ⊆r name-morph-domain(f;I)
14. bx : A-face(X;A;I;alpha) List
15. A-adjacent-compatible(X;A;I;alpha;bx)
16. l_subset(Cname;J;I)
17. ∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
18. (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
19. (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
20. (∀f∈bx.(fst(f) ∈ [x / J]))
21. (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
22. ∀x:nameset(J). (f x ∈ nameset(K))
23. map(f;J) ∈ nameset(K) List
24. f x ∈ nameset(K)
25. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
26. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
27. y : nameset(J)
28. (y ∈ J)
29. (f x) = (f y) ∈ Cname
30. J ∈ nameset(J) List
31. f ∈ nameset(J) ⟶ Cname
⊢ ↑isname(f x)
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 : nameset(I) ⟶ extd-nameset(K)
8. ∀i,j:nameset(I). ((↑isname(f i)) ⇒ (↑isname(f j)) ⇒ ((f i) = (f j) ∈ extd-nameset(K)) ⇒ (i = j ∈ nameset(I)))
9. nameset(map(f;J)) ⊆r nameset(K)
10. x : nameset(I)
11. ∀x:nameset([x / J]). (f x ∈ nameset(K))
12. i : ℕ2
13. nameset([x / J]) ⊆r name-morph-domain(f;I)
14. bx : A-face(X;A;I;alpha) List
15. A-adjacent-compatible(X;A;I;alpha;bx)
16. l_subset(Cname;J;I)
17. ∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
18. (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
19. (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
20. (∀f∈bx.(fst(f) ∈ [x / J]))
21. (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
22. ∀x:nameset(J). (f x ∈ nameset(K))
23. map(f;J) ∈ nameset(K) List
24. f x ∈ nameset(K)
25. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
26. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
27. y : nameset(J)
28. (y ∈ J)
29. (f x) = (f y) ∈ Cname
30. J ∈ nameset(J) List
31. f ∈ nameset(J) ⟶ Cname
⊢ ↑isname(f y)
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 : nameset(I) ⟶ extd-nameset(K)
8. ∀i,j:nameset(I). ((↑isname(f i)) ⇒ (↑isname(f j)) ⇒ ((f i) = (f j) ∈ extd-nameset(K)) ⇒ (i = j ∈ nameset(I)))
9. nameset(map(f;J)) ⊆r nameset(K)
10. x : nameset(I)
11. ∀x:nameset([x / J]). (f x ∈ nameset(K))
12. i : ℕ2
13. nameset([x / J]) ⊆r name-morph-domain(f;I)
14. bx : A-face(X;A;I;alpha) List
15. A-adjacent-compatible(X;A;I;alpha;bx)
16. l_subset(Cname;J;I)
17. ∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
18. (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
19. (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
20. (∀f∈bx.(fst(f) ∈ [x / J]))
21. (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
22. ∀x:nameset(J). (f x ∈ nameset(K))
23. map(f;J) ∈ nameset(K) List
24. f x ∈ nameset(K)
25. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
26. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
27. y : nameset(J)
28. (y ∈ J)
29. (f x) = (f y) ∈ Cname
30. J ∈ nameset(J) List
31. f ∈ nameset(J) ⟶ Cname
⊢ (f x) = (f y) ∈ extd-nameset(K)
4
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)
16. ∀y:nameset(J). ∀c:ℕ2. (∃f∈bx. A-face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))
17. (∃f∈bx. A-face-name(f) = <x, i> ∈ (nameset(I) × ℕ2))
18. (∀f∈bx.¬(A-face-name(f) = <x, 1 - i> ∈ (nameset(I) × ℕ2)))
19. (∀f∈bx.(fst(f) ∈ [x / J]))
20. (∀f1,f2∈bx. ¬(A-face-name(f1) = A-face-name(f2) ∈ (nameset(I) × ℕ2)))
21. ∀x:nameset(J). (f x ∈ nameset(K))
22. map(f;J) ∈ nameset(K) List
23. f x ∈ nameset(K)
24. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (fst(fc) ∈ [x / J]))
25. ∀fc:A-face(X;A;I;alpha). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
26. y : nameset(J)
27. (y ∈ J)
28. (f x) = (f y) ∈ Cname
29. J ∈ nameset(J) List
30. f ∈ nameset(J) ⟶ Cname
31. x = y ∈ nameset(I)
⊢ (x ∈ J)
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. l\_subset(Cname;J;I)
16. \mforall{}y:nameset(J). \mforall{}c:\mBbbN{}2. (\mexists{}f\mmember{}bx. A-face-name(f) = <y, c>)
17. (\mexists{}f\mmember{}bx. A-face-name(f) = <x, i>)
18. (\mforall{}f\mmember{}bx.\mneg{}(A-face-name(f) = <x, 1 - i>))
19. (\mforall{}f\mmember{}bx.(fst(f) \mmember{} [x / J]))
20. (\mforall{}f1,f2\mmember{}bx. \mneg{}(A-face-name(f1) = A-face-name(f2)))
21. \mforall{}x:nameset(J). (f x \mmember{} nameset(K))
22. map(f;J) \mmember{} nameset(K) List
23. f x \mmember{} nameset(K)
24. \mforall{}fc:A-face(X;A;I;alpha). ((fc \mmember{} bx) {}\mRightarrow{} (fst(fc) \mmember{} [x / J]))
25. \mforall{}fc:A-face(X;A;I;alpha). ((fc \mmember{} bx) {}\mRightarrow{} (\muparrow{}isname(f (fst(fc)))))
26. y : nameset(J)
27. (y \mmember{} J)
28. (f x) = (f y)
29. J \mmember{} nameset(J) List
30. f \mmember{} nameset(J) {}\mrightarrow{} Cname
\mvdash{} (x \mmember{} J)
By
Latex:
(Assert x = y BY
(DVar `f' THEN BackThruSomeHyp))
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