Step
*
1
1
2
2
of Lemma
get_face_image
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. ∀y:nameset([x / J]). (↑isname(f y))
14. L : nameset(J) List@i
15. J = L ∈ (nameset(J) List)
16. f ∈ nameset(J) ⟶ nameset(K)
⊢ (∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc))))))
∧ (map(f;L) ∈ nameset(K) List)
∧ (f x ∈ nameset(K))
∧ (f y ∈ nameset(map(f;L)))
BY
{ TACTIC:BetterSplitAndConcl }
1
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. ∀y:nameset([x / J]). (↑isname(f y))
14. L : nameset(J) List@i
15. J = L ∈ (nameset(J) List)
16. f ∈ nameset(J) ⟶ nameset(K)
⊢ ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
2
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. ∀y:nameset([x / J]). (↑isname(f y))
14. L : nameset(J) List@i
15. J = L ∈ (nameset(J) List)
16. f ∈ nameset(J) ⟶ nameset(K)
17. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
⊢ map(f;L) ∈ nameset(K) List
3
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. ∀y:nameset([x / J]). (↑isname(f y))
14. L : nameset(J) List@i
15. J = L ∈ (nameset(J) List)
16. f ∈ nameset(J) ⟶ nameset(K)
17. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
18. map(f;L) ∈ nameset(K) List
⊢ f x ∈ nameset(K)
4
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : ℕ2
6. bx : open_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : ℕ2
10. y : nameset(J)
11. nameset([x / J]) ⊆r name-morph-domain(f;I)
12. ¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List))
13. ∀y:nameset([x / J]). (↑isname(f y))
14. L : nameset(J) List@i
15. J = L ∈ (nameset(J) List)
16. f ∈ nameset(J) ⟶ nameset(K)
17. ∀fc:I-face(X;I). ((fc ∈ bx) ⇒ (↑isname(f (fst(fc)))))
18. map(f;L) ∈ nameset(K) List
19. f x ∈ nameset(K)
⊢ f y ∈ nameset(map(f;L))
Latex:
Latex:
1. X : CubicalSet
2. I : Cname List
3. J : nameset(I) List
4. x : nameset(I)
5. i : \mBbbN{}2
6. bx : open\_box(X;I;J;x;i)
7. K : Cname List
8. f : name-morph(I;K)
9. c : \mBbbN{}2
10. y : nameset(J)
11. nameset([x / J]) \msubseteq{}r name-morph-domain(f;I)
12. \mneg{}(filter(\mlambda{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx) = [])
13. \mforall{}y:nameset([x / J]). (\muparrow{}isname(f y))
14. L : nameset(J) List@i
15. J = L
16. f \mmember{} nameset(J) {}\mrightarrow{} nameset(K)
\mvdash{} (\mforall{}fc:I-face(X;I). ((fc \mmember{} bx) {}\mRightarrow{} (\muparrow{}isname(f (fst(fc))))))
\mwedge{} (map(f;L) \mmember{} nameset(K) List)
\mwedge{} (f x \mmember{} nameset(K))
\mwedge{} (f y \mmember{} nameset(map(f;L)))
By
Latex:
TACTIC:BetterSplitAndConcl
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