Proof of Theorem poimirlem23
Step | Hyp | Ref
| Expression |
1 | | ovex 6842 |
. . . . . 6
⊢ (𝑇 ∘𝑓 +
(((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V |
2 | 1 | csbex 4945 |
. . . . 5
⊢
⦋if(𝑦
< 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V |
3 | 2 | rgenw 3062 |
. . . 4
⊢
∀𝑦 ∈
(0...(𝑁 −
1))⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V |
4 | | eqid 2760 |
. . . . 5
⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) |
5 | | fveq1 6352 |
. . . . . . 7
⊢ (𝑝 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝‘𝑁) = (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁)) |
6 | 5 | neeq1d 2991 |
. . . . . 6
⊢ (𝑝 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝‘𝑁) ≠ 0 ↔ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0)) |
7 | | df-ne 2933 |
. . . . . 6
⊢
((⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0 ↔ ¬
(⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) |
8 | 6, 7 | syl6bb 276 |
. . . . 5
⊢ (𝑝 = ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝‘𝑁) ≠ 0 ↔ ¬
(⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)) |
9 | 4, 8 | rexrnmpt 6533 |
. . . 4
⊢
(∀𝑦 ∈
(0...(𝑁 −
1))⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V →
(∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)) |
10 | 3, 9 | ax-mp 5 |
. . 3
⊢
(∃𝑝 ∈ ran
(𝑦 ∈ (0...(𝑁 − 1)) ↦
⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) |
11 | | rexnal 3133 |
. . 3
⊢
(∃𝑦 ∈
(0...(𝑁 − 1)) ¬
(⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) |
12 | 10, 11 | bitri 264 |
. 2
⊢
(∃𝑝 ∈ ran
(𝑦 ∈ (0...(𝑁 − 1)) ↦
⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) |
13 | | poimir.0 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℕ) |
14 | 13 | nnzd 11693 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℤ) |
15 | | poimirlem23.3 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑉 ∈ (0...𝑁)) |
16 | | elfzelz 12555 |
. . . . . . . . . . 11
⊢ (𝑉 ∈ (0...𝑁) → 𝑉 ∈ ℤ) |
17 | 15, 16 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 ∈ ℤ) |
18 | | zlem1lt 11641 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 𝑉 ∈ ℤ) → (𝑁 ≤ 𝑉 ↔ (𝑁 − 1) < 𝑉)) |
19 | 14, 17, 18 | syl2anc 696 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 ≤ 𝑉 ↔ (𝑁 − 1) < 𝑉)) |
20 | | elfzle2 12558 |
. . . . . . . . . . 11
⊢ (𝑉 ∈ (0...𝑁) → 𝑉 ≤ 𝑁) |
21 | 15, 20 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 ≤ 𝑁) |
22 | 17 | zred 11694 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑉 ∈ ℝ) |
23 | 13 | nnred 11247 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℝ) |
24 | 22, 23 | letri3d 10391 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑉 = 𝑁 ↔ (𝑉 ≤ 𝑁 ∧ 𝑁 ≤ 𝑉))) |
25 | 24 | biimprd 238 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑉 ≤ 𝑁 ∧ 𝑁 ≤ 𝑉) → 𝑉 = 𝑁)) |
26 | 21, 25 | mpand 713 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 ≤ 𝑉 → 𝑉 = 𝑁)) |
27 | 19, 26 | sylbird 250 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 − 1) < 𝑉 → 𝑉 = 𝑁)) |
28 | 27 | necon3ad 2945 |
. . . . . . 7
⊢ (𝜑 → (𝑉 ≠ 𝑁 → ¬ (𝑁 − 1) < 𝑉)) |
29 | | nnm1nn0 11546 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ0) |
30 | 13, 29 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 − 1) ∈
ℕ0) |
31 | | nn0fz0 12651 |
. . . . . . . . . . . 12
⊢ ((𝑁 − 1) ∈
ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1))) |
32 | 30, 31 | sylib 208 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1))) |
33 | 32 | adantr 472 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (𝑁 − 1) ∈ (0...(𝑁 − 1))) |
34 | | iffalse 4239 |
. . . . . . . . . . . . . . . 16
⊢ (¬
(𝑁 − 1) < 𝑉 → if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) = ((𝑁 − 1) + 1)) |
35 | 13 | nncnd 11248 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℂ) |
36 | | npcan1 10667 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
37 | 35, 36 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
38 | 34, 37 | sylan9eqr 2816 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) = 𝑁) |
39 | 38 | csbeq1d 3681 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑁 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) |
40 | | oveq2 6822 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁)) |
41 | 40 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑁 → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...𝑁))) |
42 | 41 | xpeq1d 5295 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑁 → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...𝑁)) × {1})) |
43 | | oveq1 6821 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1)) |
44 | 43 | oveq1d 6829 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁)) |
45 | 44 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑁 → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ ((𝑁 + 1)...𝑁))) |
46 | 45 | xpeq1d 5295 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑁 → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})) |
47 | 42, 46 | uneq12d 3911 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑁 → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}))) |
48 | | poimirlem23.2 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)) |
49 | | f1ofo 6306 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁)) |
50 | | foima 6282 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁)) |
51 | 48, 49, 50 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁)) |
52 | 51 | xpeq1d 5295 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑈 “ (1...𝑁)) × {1}) = ((1...𝑁) × {1})) |
53 | 23 | ltp1d 11166 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 < (𝑁 + 1)) |
54 | 14 | peano2zd 11697 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
55 | | fzn 12570 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅)) |
56 | 54, 14, 55 | syl2anc 696 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅)) |
57 | 53, 56 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅) |
58 | 57 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑈 “ ((𝑁 + 1)...𝑁)) = (𝑈 “ ∅)) |
59 | 58 | xpeq1d 5295 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}) = ((𝑈 “ ∅) ×
{0})) |
60 | | ima0 5639 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑈 “ ∅) =
∅ |
61 | 60 | xpeq1i 5292 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑈 “ ∅) × {0}) =
(∅ × {0}) |
62 | | 0xp 5356 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (∅
× {0}) = ∅ |
63 | 61, 62 | eqtri 2782 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑈 “ ∅) × {0}) =
∅ |
64 | 59, 63 | syl6eq 2810 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}) = ∅) |
65 | 52, 64 | uneq12d 3911 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪
∅)) |
66 | | un0 4110 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...𝑁) ×
{1}) ∪ ∅) = ((1...𝑁) × {1}) |
67 | 65, 66 | syl6eq 2810 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1})) |
68 | 47, 67 | sylan9eqr 2816 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1})) |
69 | 68 | oveq2d 6830 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 = 𝑁) → (𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + ((1...𝑁) ×
{1}))) |
70 | 13, 69 | csbied 3701 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ⦋𝑁 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + ((1...𝑁) ×
{1}))) |
71 | 70 | adantr 472 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ⦋𝑁 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + ((1...𝑁) ×
{1}))) |
72 | 39, 71 | eqtrd 2794 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + ((1...𝑁) ×
{1}))) |
73 | 72 | fveq1d 6355 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇 ∘𝑓 + ((1...𝑁) × {1}))‘𝑁)) |
74 | | elfzonn0 12727 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0..^𝐾) → 𝑗 ∈ ℕ0) |
75 | | nn0p1nn 11544 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ ℕ0
→ (𝑗 + 1) ∈
ℕ) |
76 | 74, 75 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (0..^𝐾) → (𝑗 + 1) ∈ ℕ) |
77 | | elsni 4338 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ {1} → 𝑦 = 1) |
78 | 77 | oveq2d 6830 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ {1} → (𝑗 + 𝑦) = (𝑗 + 1)) |
79 | 78 | eleq1d 2824 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ {1} → ((𝑗 + 𝑦) ∈ ℕ ↔ (𝑗 + 1) ∈ ℕ)) |
80 | 76, 79 | syl5ibrcom 237 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (0..^𝐾) → (𝑦 ∈ {1} → (𝑗 + 𝑦) ∈ ℕ)) |
81 | 80 | imp 444 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑗 ∈ (0..^𝐾) ∧ 𝑦 ∈ {1}) → (𝑗 + 𝑦) ∈ ℕ) |
82 | 81 | adantl 473 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑗 ∈ (0..^𝐾) ∧ 𝑦 ∈ {1})) → (𝑗 + 𝑦) ∈ ℕ) |
83 | | poimirlem23.1 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾)) |
84 | | 1ex 10247 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
V |
85 | 84 | fconst 6252 |
. . . . . . . . . . . . . . . 16
⊢
((1...𝑁) ×
{1}):(1...𝑁)⟶{1} |
86 | 85 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((1...𝑁) × {1}):(1...𝑁)⟶{1}) |
87 | | ovexd 6844 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (1...𝑁) ∈ V) |
88 | | inidm 3965 |
. . . . . . . . . . . . . . 15
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
89 | 82, 83, 86, 87, 87, 88 | off 7078 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑇 ∘𝑓 + ((1...𝑁) × {1})):(1...𝑁)⟶ℕ) |
90 | | elfz1end 12584 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁)) |
91 | 13, 90 | sylib 208 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ (1...𝑁)) |
92 | 89, 91 | ffvelrnd 6524 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑇 ∘𝑓 + ((1...𝑁) × {1}))‘𝑁) ∈
ℕ) |
93 | 92 | adantr 472 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ((𝑇 ∘𝑓 + ((1...𝑁) × {1}))‘𝑁) ∈
ℕ) |
94 | 73, 93 | eqeltrd 2839 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ∈ ℕ) |
95 | 94 | nnne0d 11277 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0) |
96 | | breq1 4807 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = (𝑁 − 1) → (𝑦 < 𝑉 ↔ (𝑁 − 1) < 𝑉)) |
97 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = (𝑁 − 1) → 𝑦 = (𝑁 − 1)) |
98 | | oveq1 6821 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1)) |
99 | 96, 97, 98 | ifbieq12d 4257 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑁 − 1) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1))) |
100 | 99 | csbeq1d 3681 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑁 − 1) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) |
101 | 100 | fveq1d 6355 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑁 − 1) → (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁)) |
102 | 101 | neeq1d 2991 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑁 − 1) → ((⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0 ↔ (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0)) |
103 | 7, 102 | syl5bbr 274 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑁 − 1) → (¬
(⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (⦋if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0)) |
104 | 103 | rspcev 3449 |
. . . . . . . . . 10
⊢ (((𝑁 − 1) ∈ (0...(𝑁 − 1)) ∧
(⦋if((𝑁
− 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0) → ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) |
105 | 33, 95, 104 | syl2anc 696 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) |
106 | 105, 11 | sylib 208 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) |
107 | 106 | ex 449 |
. . . . . . 7
⊢ (𝜑 → (¬ (𝑁 − 1) < 𝑉 → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)) |
108 | 28, 107 | syld 47 |
. . . . . 6
⊢ (𝜑 → (𝑉 ≠ 𝑁 → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)) |
109 | 108 | necon4ad 2951 |
. . . . 5
⊢ (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 → 𝑉 = 𝑁)) |
110 | 109 | pm4.71rd 670 |
. . . 4
⊢ (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (𝑉 = 𝑁 ∧ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))) |
111 | 30 | nn0zd 11692 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
112 | | uzid 11914 |
. . . . . . . . . . . . 13
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) |
113 | | peano2uz 11954 |
. . . . . . . . . . . . 13
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
114 | 111, 112,
113 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
115 | 37, 114 | eqeltrrd 2840 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
116 | | fzss2 12594 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁)) |
117 | 115, 116 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁)) |
118 | 117 | sselda 3744 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ (0...𝑁)) |
119 | 91 | adantr 472 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ∈ (1...𝑁)) |
120 | | ffn 6206 |
. . . . . . . . . . . . . . 15
⊢ (𝑇:(1...𝑁)⟶(0..^𝐾) → 𝑇 Fn (1...𝑁)) |
121 | 83, 120 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 Fn (1...𝑁)) |
122 | 121 | adantr 472 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑇 Fn (1...𝑁)) |
123 | 84 | fconst 6252 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} |
124 | | c0ex 10246 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
V |
125 | 124 | fconst 6252 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0} |
126 | 123, 125 | pm3.2i 470 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}) |
127 | | dff1o3 6305 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡𝑈)) |
128 | 127 | simprbi 483 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡𝑈) |
129 | | imain 6135 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
◡𝑈 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁)))) |
130 | 48, 128, 129 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁)))) |
131 | | elfzelz 12555 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ) |
132 | 131 | zred 11694 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ) |
133 | 132 | ltp1d 11166 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1)) |
134 | | fzdisj 12581 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅) |
135 | 133, 134 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅) |
136 | 135 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (𝑈 “ ∅)) |
137 | 136, 60 | syl6eq 2810 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅) |
138 | 130, 137 | sylan9req 2815 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅) |
139 | | fun 6227 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑈 “
(1...𝑗)) ×
{1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0})) |
140 | 126, 138,
139 | sylancr 698 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0})) |
141 | | elfznn0 12646 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ0) |
142 | 141, 75 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ) |
143 | | nnuz 11936 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ℕ =
(ℤ≥‘1) |
144 | 142, 143 | syl6eleq 2849 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈
(ℤ≥‘1)) |
145 | | elfzuz3 12552 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ≥‘𝑗)) |
146 | | fzsplit2 12579 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑗 + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) |
147 | 144, 145,
146 | syl2anc 696 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) |
148 | 147 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (0...𝑁) → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))) |
149 | | imaundi 5703 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) |
150 | 148, 149 | syl6req 2811 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (0...𝑁) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (𝑈 “ (1...𝑁))) |
151 | 150, 51 | sylan9eqr 2816 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (1...𝑁)) |
152 | 151 | feq2d 6192 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))) |
153 | 140, 152 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})) |
154 | | ffn 6206 |
. . . . . . . . . . . . . 14
⊢ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
155 | 153, 154 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
156 | | ovexd 6844 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ V) |
157 | | eqidd 2761 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → (𝑇‘𝑁) = (𝑇‘𝑁)) |
158 | | eqidd 2761 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) |
159 | 122, 155,
156, 156, 88, 157, 158 | ofval 7072 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → ((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))) |
160 | 119, 159 | mpdan 705 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))) |
161 | 160 | eqeq1d 2762 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0)) |
162 | 83, 91 | ffvelrnd 6524 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑇‘𝑁) ∈ (0..^𝐾)) |
163 | | elfzonn0 12727 |
. . . . . . . . . . . . . 14
⊢ ((𝑇‘𝑁) ∈ (0..^𝐾) → (𝑇‘𝑁) ∈
ℕ0) |
164 | 162, 163 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑇‘𝑁) ∈
ℕ0) |
165 | 164 | nn0red 11564 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑇‘𝑁) ∈ ℝ) |
166 | 165 | adantr 472 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑇‘𝑁) ∈ ℝ) |
167 | 164 | nn0ge0d 11566 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤ (𝑇‘𝑁)) |
168 | 167 | adantr 472 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ≤ (𝑇‘𝑁)) |
169 | | 1re 10251 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℝ |
170 | | snssi 4484 |
. . . . . . . . . . . . . 14
⊢ (1 ∈
ℝ → {1} ⊆ ℝ) |
171 | 169, 170 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ {1}
⊆ ℝ |
172 | | 0re 10252 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ |
173 | | snssi 4484 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
ℝ → {0} ⊆ ℝ) |
174 | 172, 173 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ {0}
⊆ ℝ |
175 | 171, 174 | unssi 3931 |
. . . . . . . . . . . 12
⊢ ({1}
∪ {0}) ⊆ ℝ |
176 | 153, 119 | ffvelrnd 6524 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0})) |
177 | 175, 176 | sseldi 3742 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ℝ) |
178 | | elun 3896 |
. . . . . . . . . . . . 13
⊢
(((((𝑈 “
(1...𝑗)) × {1}) ∪
((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0}) ↔ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} ∨ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0})) |
179 | | 0le1 10763 |
. . . . . . . . . . . . . . 15
⊢ 0 ≤
1 |
180 | | elsni 4338 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑈 “
(1...𝑗)) × {1}) ∪
((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 1) |
181 | 179, 180 | syl5breqr 4842 |
. . . . . . . . . . . . . 14
⊢
(((((𝑈 “
(1...𝑗)) × {1}) ∪
((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) |
182 | | 0le0 11322 |
. . . . . . . . . . . . . . 15
⊢ 0 ≤
0 |
183 | | elsni 4338 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑈 “
(1...𝑗)) × {1}) ∪
((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0} → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) |
184 | 182, 183 | syl5breqr 4842 |
. . . . . . . . . . . . . 14
⊢
(((((𝑈 “
(1...𝑗)) × {1}) ∪
((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0} → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) |
185 | 181, 184 | jaoi 393 |
. . . . . . . . . . . . 13
⊢
((((((𝑈 “
(1...𝑗)) × {1}) ∪
((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} ∨ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0}) → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) |
186 | 178, 185 | sylbi 207 |
. . . . . . . . . . . 12
⊢
(((((𝑈 “
(1...𝑗)) × {1}) ∪
((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0}) → 0 ≤
((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) |
187 | 176, 186 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) |
188 | | add20 10752 |
. . . . . . . . . . 11
⊢ ((((𝑇‘𝑁) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑁)) ∧ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ℝ ∧ 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))) → (((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
189 | 166, 168,
177, 187, 188 | syl22anc 1478 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑇‘𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
190 | 161, 189 | bitrd 268 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
191 | 118, 190 | syldan 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
192 | 191 | ralbidva 3123 |
. . . . . . 7
⊢ (𝜑 → (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
193 | 192 | adantr 472 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
194 | | breq2 4808 |
. . . . . . . . . . . . . 14
⊢ (𝑉 = 𝑁 → (𝑦 < 𝑉 ↔ 𝑦 < 𝑁)) |
195 | 194 | ifbid 4252 |
. . . . . . . . . . . . 13
⊢ (𝑉 = 𝑁 → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = if(𝑦 < 𝑁, 𝑦, (𝑦 + 1))) |
196 | | elfzelz 12555 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ) |
197 | 196 | zred 11694 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ) |
198 | 197 | adantl 473 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ) |
199 | 30 | nn0red 11564 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
200 | 199 | adantr 472 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ) |
201 | 23 | adantr 472 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ) |
202 | | elfzle2 12558 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1)) |
203 | 202 | adantl 473 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1)) |
204 | 23 | ltm1d 11168 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
205 | 204 | adantr 472 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁) |
206 | 198, 200,
201, 203, 205 | lelttrd 10407 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁) |
207 | 206 | iftrued 4238 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑁, 𝑦, (𝑦 + 1)) = 𝑦) |
208 | 195, 207 | sylan9eqr 2816 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑉 = 𝑁) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = 𝑦) |
209 | 208 | an32s 881 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = 𝑦) |
210 | 209 | csbeq1d 3681 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) |
211 | 210 | fveq1d 6355 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) →
(⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁)) |
212 | 211 | eqeq1d 2762 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) →
((⦋if(𝑦 <
𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)) |
213 | 212 | ralbidva 3123 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)) |
214 | | nfv 1992 |
. . . . . . . 8
⊢
Ⅎ𝑦((𝑇 ∘𝑓 +
(((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 |
215 | | nfcsb1v 3690 |
. . . . . . . . . 10
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) |
216 | | nfcv 2902 |
. . . . . . . . . 10
⊢
Ⅎ𝑗𝑁 |
217 | 215, 216 | nffv 6360 |
. . . . . . . . 9
⊢
Ⅎ𝑗(⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) |
218 | 217 | nfeq1 2916 |
. . . . . . . 8
⊢
Ⅎ𝑗(⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 |
219 | | csbeq1a 3683 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → (𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) |
220 | 219 | fveq1d 6355 |
. . . . . . . . 9
⊢ (𝑗 = 𝑦 → ((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁)) |
221 | 220 | eqeq1d 2762 |
. . . . . . . 8
⊢ (𝑗 = 𝑦 → (((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)) |
222 | 214, 218,
221 | cbvral 3306 |
. . . . . . 7
⊢
(∀𝑗 ∈
(0...(𝑁 − 1))((𝑇 ∘𝑓 +
(((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋𝑦 / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) |
223 | 213, 222 | syl6bbr 278 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)) |
224 | | ne0i 4064 |
. . . . . . . . . 10
⊢ ((𝑁 − 1) ∈ (0...(𝑁 − 1)) → (0...(𝑁 − 1)) ≠
∅) |
225 | | r19.3rzv 4208 |
. . . . . . . . . 10
⊢
((0...(𝑁 − 1))
≠ ∅ → ((𝑇‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))(𝑇‘𝑁) = 0)) |
226 | 32, 224, 225 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑇‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))(𝑇‘𝑁) = 0)) |
227 | | elfzelz 12555 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℤ) |
228 | 227 | zred 11694 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℝ) |
229 | 228 | ltp1d 11166 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 < (𝑗 + 1)) |
230 | 229, 134 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅) |
231 | 230 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (𝑈 “ ∅)) |
232 | 231, 60 | syl6eq 2810 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅) |
233 | 130, 232 | sylan9req 2815 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅) |
234 | 233 | adantlr 753 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅) |
235 | | simplr 809 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑈‘𝑁) = 𝑁) |
236 | | f1ofn 6300 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁)) |
237 | 48, 236 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑈 Fn (1...𝑁)) |
238 | 237 | adantr 472 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑈 Fn (1...𝑁)) |
239 | | elfznn0 12646 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℕ0) |
240 | 239, 75 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈ ℕ) |
241 | 240, 143 | syl6eleq 2849 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈
(ℤ≥‘1)) |
242 | | fzss1 12593 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑗 + 1) ∈
(ℤ≥‘1) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁)) |
243 | 241, 242 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁)) |
244 | 243 | adantl 473 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁)) |
245 | 37 | adantr 472 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
246 | | elfzuz3 12552 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ≥‘𝑗)) |
247 | | eluzp1p1 11925 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑗) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑗 + 1))) |
248 | 246, 247 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑗 + 1))) |
249 | 248 | adantl 473 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑗 + 1))) |
250 | 245, 249 | eqeltrrd 2840 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ≥‘(𝑗 + 1))) |
251 | | eluzfz2 12562 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈
(ℤ≥‘(𝑗 + 1)) → 𝑁 ∈ ((𝑗 + 1)...𝑁)) |
252 | 250, 251 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑗 + 1)...𝑁)) |
253 | | fnfvima 6660 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑈 Fn (1...𝑁) ∧ ((𝑗 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑗 + 1)...𝑁)) → (𝑈‘𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁))) |
254 | 238, 244,
252, 253 | syl3anc 1477 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑈‘𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁))) |
255 | 254 | adantlr 753 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑈‘𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁))) |
256 | 235, 255 | eqeltrrd 2840 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁))) |
257 | | fnconstg 6254 |
. . . . . . . . . . . . . . . 16
⊢ (1 ∈
V → ((𝑈 “
(1...𝑗)) × {1}) Fn
(𝑈 “ (1...𝑗))) |
258 | 84, 257 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) |
259 | | fnconstg 6254 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
V → ((𝑈 “
((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁))) |
260 | 124, 259 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁)) |
261 | | fvun2 6433 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁)) ∧ (((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁)) |
262 | 258, 260,
261 | mp3an12 1563 |
. . . . . . . . . . . . . 14
⊢ ((((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁)) |
263 | 234, 256,
262 | syl2anc 696 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁)) |
264 | 124 | fvconst2 6634 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁)) → (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁) = 0) |
265 | 256, 264 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁) = 0) |
266 | 263, 265 | eqtrd 2794 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑈‘𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) |
267 | 266 | ralrimiva 3104 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑈‘𝑁) = 𝑁) → ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) |
268 | 267 | ex 449 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑈‘𝑁) = 𝑁 → ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)) |
269 | 32 | adantr 472 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → (𝑁 − 1) ∈ (0...(𝑁 − 1))) |
270 | | ax-1ne0 10217 |
. . . . . . . . . . . . . . 15
⊢ 1 ≠
0 |
271 | | imain 6135 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (Fun
◡𝑈 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁)))) |
272 | 48, 128, 271 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁)))) |
273 | 204, 37 | breqtrrd 4832 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑁 − 1) < ((𝑁 − 1) + 1)) |
274 | | fzdisj 12581 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑁 − 1) < ((𝑁 − 1) + 1) →
((1...(𝑁 − 1)) ∩
(((𝑁 − 1) +
1)...𝑁)) =
∅) |
275 | 273, 274 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁)) = ∅) |
276 | 275 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = (𝑈 “ ∅)) |
277 | 276, 60 | syl6eq 2810 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ∅) |
278 | 272, 277 | eqtr3d 2796 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅) |
279 | 278 | adantr 472 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅) |
280 | 91 | adantr 472 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → 𝑁 ∈ (1...𝑁)) |
281 | | elimasni 5650 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈ (𝑈 “ {𝑁}) → 𝑁𝑈𝑁) |
282 | | fnbrfvb 6398 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑈 Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → ((𝑈‘𝑁) = 𝑁 ↔ 𝑁𝑈𝑁)) |
283 | 237, 91, 282 | syl2anc 696 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑈‘𝑁) = 𝑁 ↔ 𝑁𝑈𝑁)) |
284 | 281, 283 | syl5ibr 236 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑁 ∈ (𝑈 “ {𝑁}) → (𝑈‘𝑁) = 𝑁)) |
285 | 284 | necon3ad 2945 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑈‘𝑁) ≠ 𝑁 → ¬ 𝑁 ∈ (𝑈 “ {𝑁}))) |
286 | 285 | imp 444 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ¬ 𝑁 ∈ (𝑈 “ {𝑁})) |
287 | 280, 286 | eldifd 3726 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → 𝑁 ∈ ((1...𝑁) ∖ (𝑈 “ {𝑁}))) |
288 | | imadif 6134 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (Fun
◡𝑈 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁}))) |
289 | 48, 128, 288 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁}))) |
290 | | difun2 4192 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1...(𝑁 −
1)) ∪ {𝑁}) ∖
{𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁}) |
291 | 13, 143 | syl6eleq 2849 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
292 | | fzm1 12633 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑁 ∈
(ℤ≥‘1) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁))) |
293 | 291, 292 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁))) |
294 | | elun 3896 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 ∈ {𝑁})) |
295 | | velsn 4337 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ {𝑁} ↔ 𝑗 = 𝑁) |
296 | 295 | orbi2i 542 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 ∈ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁)) |
297 | 294, 296 | bitri 264 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁)) |
298 | 293, 297 | syl6rbbr 279 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ 𝑗 ∈ (1...𝑁))) |
299 | 298 | eqrdv 2758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁)) |
300 | 299 | difeq1d 3870 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...𝑁) ∖ {𝑁})) |
301 | 199, 23 | ltnled 10396 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1))) |
302 | 204, 301 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1)) |
303 | | elfzle2 12558 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1)) |
304 | 302, 303 | nsyl 135 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1))) |
305 | | difsn 4473 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (¬
𝑁 ∈ (1...(𝑁 − 1)) → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1))) |
306 | 304, 305 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1))) |
307 | 290, 300,
306 | 3eqtr3a 2818 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1))) |
308 | 307 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = (𝑈 “ (1...(𝑁 − 1)))) |
309 | 51 | difeq1d 3870 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁})) = ((1...𝑁) ∖ (𝑈 “ {𝑁}))) |
310 | 289, 308,
309 | 3eqtr3rd 2803 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((1...𝑁) ∖ (𝑈 “ {𝑁})) = (𝑈 “ (1...(𝑁 − 1)))) |
311 | 310 | adantr 472 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((1...𝑁) ∖ (𝑈 “ {𝑁})) = (𝑈 “ (1...(𝑁 − 1)))) |
312 | 287, 311 | eleqtrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1)))) |
313 | | fnconstg 6254 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (1 ∈
V → ((𝑈 “
(1...(𝑁 − 1)))
× {1}) Fn (𝑈 “
(1...(𝑁 −
1)))) |
314 | 84, 313 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑈 “ (1...(𝑁 − 1))) × {1}) Fn (𝑈 “ (1...(𝑁 − 1))) |
315 | | fnconstg 6254 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 ∈
V → ((𝑈 “
(((𝑁 − 1) +
1)...𝑁)) × {0}) Fn
(𝑈 “ (((𝑁 − 1) + 1)...𝑁))) |
316 | 124, 315 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑁 − 1) + 1)...𝑁)) |
317 | | fvun1 6432 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) Fn (𝑈 “ (1...(𝑁 − 1))) ∧ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑁 − 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1))))) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁)) |
318 | 314, 316,
317 | mp3an12 1563 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1)))) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁)) |
319 | 279, 312,
318 | syl2anc 696 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁)) |
320 | 84 | fvconst2 6634 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ (𝑈 “ (1...(𝑁 − 1))) → (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁) = 1) |
321 | 312, 320 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁) = 1) |
322 | 319, 321 | eqtrd 2794 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = 1) |
323 | 322 | neeq1d 2991 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → (((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ 1 ≠ 0)) |
324 | 270, 323 | mpbiri 248 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0) |
325 | | df-ne 2933 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑈 “
(1...𝑗)) × {1}) ∪
((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) |
326 | | oveq2 6822 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑁 − 1) → (1...𝑗) = (1...(𝑁 − 1))) |
327 | 326 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑁 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑁 − 1)))) |
328 | 327 | xpeq1d 5295 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑁 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑁 − 1))) × {1})) |
329 | | oveq1 6821 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = (𝑁 − 1) → (𝑗 + 1) = ((𝑁 − 1) + 1)) |
330 | 329 | oveq1d 6829 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = (𝑁 − 1) → ((𝑗 + 1)...𝑁) = (((𝑁 − 1) + 1)...𝑁)) |
331 | 330 | imaeq2d 5624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑁 − 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) |
332 | 331 | xpeq1d 5295 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑁 − 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0})) |
333 | 328, 332 | uneq12d 3911 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑁 − 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))) |
334 | 333 | fveq1d 6355 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑁 − 1) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁)) |
335 | 334 | neeq1d 2991 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑁 − 1) → (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0)) |
336 | 325, 335 | syl5bbr 274 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑁 − 1) → (¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 ↔ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0)) |
337 | 336 | rspcev 3449 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 − 1) ∈ (0...(𝑁 − 1)) ∧ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0) → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) |
338 | 269, 324,
337 | syl2anc 696 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑈‘𝑁) ≠ 𝑁) → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) |
339 | 338 | ex 449 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑈‘𝑁) ≠ 𝑁 → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)) |
340 | | rexnal 3133 |
. . . . . . . . . . . 12
⊢
(∃𝑗 ∈
(0...(𝑁 − 1)) ¬
((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 ↔ ¬ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) |
341 | 339, 340 | syl6ib 241 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑈‘𝑁) ≠ 𝑁 → ¬ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)) |
342 | 341 | necon4ad 2951 |
. . . . . . . . . 10
⊢ (𝜑 → (∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 → (𝑈‘𝑁) = 𝑁)) |
343 | 268, 342 | impbid 202 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑈‘𝑁) = 𝑁 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)) |
344 | 226, 343 | anbi12d 749 |
. . . . . . . 8
⊢ (𝜑 → (((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁) ↔ (∀𝑗 ∈ (0...(𝑁 − 1))(𝑇‘𝑁) = 0 ∧ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
345 | | r19.26 3202 |
. . . . . . . 8
⊢
(∀𝑗 ∈
(0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) ↔ (∀𝑗 ∈ (0...(𝑁 − 1))(𝑇‘𝑁) = 0 ∧ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)) |
346 | 344, 345 | syl6bbr 278 |
. . . . . . 7
⊢ (𝜑 → (((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁) ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
347 | 346 | adantr 472 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁) ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇‘𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))) |
348 | 193, 223,
347 | 3bitr4d 300 |
. . . . 5
⊢ ((𝜑 ∧ 𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))) |
349 | 348 | pm5.32da 676 |
. . . 4
⊢ (𝜑 → ((𝑉 = 𝑁 ∧ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) ↔ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))) |
350 | 110, 349 | bitrd 268 |
. . 3
⊢ (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))) |
351 | 350 | notbid 307 |
. 2
⊢ (𝜑 → (¬ ∀𝑦 ∈ (0...(𝑁 − 1))(⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))) |
352 | 12, 351 | syl5bb 272 |
1
⊢ (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))) |