Step | Hyp | Ref
| Expression |
1 | | fzfi 13090 |
. . . . . 6
⊢
(0...𝐾) ∈
Fin |
2 | | fzfi 13090 |
. . . . . 6
⊢
(1...𝑁) ∈
Fin |
3 | | mapfi 8550 |
. . . . . 6
⊢
(((0...𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0...𝐾)
↑𝑚 (1...𝑁)) ∈ Fin) |
4 | 1, 2, 3 | mp2an 682 |
. . . . 5
⊢
((0...𝐾)
↑𝑚 (1...𝑁)) ∈ Fin |
5 | | fzfi 13090 |
. . . . 5
⊢
(0...(𝑁 − 1))
∈ Fin |
6 | | mapfi 8550 |
. . . . 5
⊢
((((0...𝐾)
↑𝑚 (1...𝑁)) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → (((0...𝐾) ↑𝑚
(1...𝑁))
↑𝑚 (0...(𝑁 − 1))) ∈ Fin) |
7 | 4, 5, 6 | mp2an 682 |
. . . 4
⊢
(((0...𝐾)
↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) ∈
Fin |
8 | 7 | a1i 11 |
. . 3
⊢ (𝜑 → (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))) ∈
Fin) |
9 | | 2z 11761 |
. . . 4
⊢ 2 ∈
ℤ |
10 | 9 | a1i 11 |
. . 3
⊢ (𝜑 → 2 ∈
ℤ) |
11 | | fzofi 13092 |
. . . . . . . 8
⊢
(0..^𝐾) ∈
Fin |
12 | | mapfi 8550 |
. . . . . . . 8
⊢
(((0..^𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0..^𝐾)
↑𝑚 (1...𝑁)) ∈ Fin) |
13 | 11, 2, 12 | mp2an 682 |
. . . . . . 7
⊢
((0..^𝐾)
↑𝑚 (1...𝑁)) ∈ Fin |
14 | | mapfi 8550 |
. . . . . . . . 9
⊢
(((1...𝑁) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((1...𝑁)
↑𝑚 (1...𝑁)) ∈ Fin) |
15 | 2, 2, 14 | mp2an 682 |
. . . . . . . 8
⊢
((1...𝑁)
↑𝑚 (1...𝑁)) ∈ Fin |
16 | | f1of 6391 |
. . . . . . . . . 10
⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁)) |
17 | 16 | ss2abi 3895 |
. . . . . . . . 9
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} |
18 | | ovex 6954 |
. . . . . . . . . 10
⊢
(1...𝑁) ∈
V |
19 | 18, 18 | mapval 8152 |
. . . . . . . . 9
⊢
((1...𝑁)
↑𝑚 (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} |
20 | 17, 19 | sseqtr4i 3857 |
. . . . . . . 8
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁)) |
21 | | ssfi 8468 |
. . . . . . . 8
⊢
((((1...𝑁)
↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁))) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) |
22 | 15, 20, 21 | mp2an 682 |
. . . . . . 7
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin |
23 | | xpfi 8519 |
. . . . . . 7
⊢
((((0..^𝐾)
↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin) |
24 | 13, 22, 23 | mp2an 682 |
. . . . . 6
⊢
(((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin |
25 | | fzfi 13090 |
. . . . . 6
⊢
(0...𝑁) ∈
Fin |
26 | | xpfi 8519 |
. . . . . 6
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin) |
27 | 24, 25, 26 | mp2an 682 |
. . . . 5
⊢
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin |
28 | | rabfi 8473 |
. . . . 5
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin) |
29 | 27, 28 | ax-mp 5 |
. . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin |
30 | | hashcl 13462 |
. . . . 5
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈
ℕ0) |
31 | 30 | nn0zd 11832 |
. . . 4
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ) |
32 | 29, 31 | mp1i 13 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (♯‘{𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ) |
33 | | dfrex2 3177 |
. . . . 5
⊢
(∃𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ↔ ¬ ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
34 | | nfv 1957 |
. . . . . 6
⊢
Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))) |
35 | | nfcv 2934 |
. . . . . . 7
⊢
Ⅎ𝑡2 |
36 | | nfcv 2934 |
. . . . . . 7
⊢
Ⅎ𝑡
∥ |
37 | | nfcv 2934 |
. . . . . . . 8
⊢
Ⅎ𝑡♯ |
38 | | nfrab1 3309 |
. . . . . . . 8
⊢
Ⅎ𝑡{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} |
39 | 37, 38 | nffv 6456 |
. . . . . . 7
⊢
Ⅎ𝑡(♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
40 | 35, 36, 39 | nfbr 4933 |
. . . . . 6
⊢
Ⅎ𝑡2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
41 | | neq0 4158 |
. . . . . . . . . . . 12
⊢ (¬
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ ↔
∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) |
42 | | iddvds 15402 |
. . . . . . . . . . . . . . . . 17
⊢ (2 ∈
ℤ → 2 ∥ 2) |
43 | 9, 42 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∥
2 |
44 | | vex 3401 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑠 ∈ V |
45 | | hashsng 13474 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ V →
(♯‘{𝑠}) =
1) |
46 | 44, 45 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢
(♯‘{𝑠})
= 1 |
47 | 46 | oveq2i 6933 |
. . . . . . . . . . . . . . . . 17
⊢ (1 +
(♯‘{𝑠})) = (1 +
1) |
48 | | df-2 11438 |
. . . . . . . . . . . . . . . . 17
⊢ 2 = (1 +
1) |
49 | 47, 48 | eqtr4i 2805 |
. . . . . . . . . . . . . . . 16
⊢ (1 +
(♯‘{𝑠})) =
2 |
50 | 43, 49 | breqtrri 4913 |
. . . . . . . . . . . . . . 15
⊢ 2 ∥
(1 + (♯‘{𝑠})) |
51 | | rabfi 8473 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈
Fin) |
52 | | diffi 8480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈ Fin →
({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin) |
53 | 27, 51, 52 | mp2b 10 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin |
54 | | snfi 8326 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑠} ∈ Fin |
55 | | incom 4028 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) |
56 | | disjdif 4264 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = ∅ |
57 | 55, 56 | eqtri 2802 |
. . . . . . . . . . . . . . . . . . 19
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅ |
58 | | hashun 13486 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin ∧ {𝑠} ∈ Fin ∧ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅) → (♯‘(({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠}))) |
59 | 53, 54, 57, 58 | mp3an 1534 |
. . . . . . . . . . . . . . . . . 18
⊢
(♯‘(({𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) |
60 | | difsnid 4572 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠}) = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) |
61 | 60 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} →
(♯‘(({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
62 | 59, 61 | syl5eqr 2828 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} →
((♯‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
63 | 62 | adantl 475 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
((♯‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
64 | | poimir.0 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℕ) |
65 | 64 | ad3antrrr 720 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑁 ∈
ℕ) |
66 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (2nd ‘𝑡) = (2nd ‘𝑢)) |
67 | 66 | breq2d 4898 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑢))) |
68 | 67 | ifbid 4329 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 𝑢 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1))) |
69 | 68 | csbeq1d 3758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
70 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑢))) |
71 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑡 = 𝑢 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑢))) |
72 | 71 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 𝑢 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑢)) “
(1...𝑗))) |
73 | 72 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1})) |
74 | 71 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 𝑢 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑢)) “ ((𝑗 + 1)...𝑁))) |
75 | 74 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) |
76 | 73, 75 | uneq12d 3991 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
77 | 70, 76 | oveq12d 6940 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 𝑢 → ((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
78 | 77 | csbeq2dv 4217 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
79 | 69, 78 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
80 | 79 | mpteq2dv 4980 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
81 | | breq1 4889 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 < (2nd ‘𝑢) ↔ 𝑤 < (2nd ‘𝑢))) |
82 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤) |
83 | | oveq1 6929 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1)) |
84 | 81, 82, 83 | ifbieq12d 4334 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = 𝑤 → if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1))) |
85 | 84 | csbeq1d 3758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
86 | | oveq2 6930 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖)) |
87 | 86 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 𝑖 → ((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑢)) “
(1...𝑖))) |
88 | 87 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 𝑖 → (((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1})) |
89 | | oveq1 6929 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1)) |
90 | 89 | oveq1d 6937 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁)) |
91 | 90 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 𝑖 → ((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑢)) “ ((𝑖 + 1)...𝑁))) |
92 | 91 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 𝑖 → (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})) |
93 | 88, 92 | uneq12d 3991 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = 𝑖 → ((((2nd
‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))) |
94 | 93 | oveq2d 6938 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 𝑖 → ((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) |
95 | 94 | cbvcsbv 3757 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
⦋if(𝑤
< (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))) |
96 | 85, 95 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) |
97 | 96 | cbvmptv 4985 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) |
98 | 80, 97 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))) |
99 | 98 | eqeq2d 2788 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑢 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))))) |
100 | 99 | cbvrabv 3396 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑢 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st
‘(1st ‘𝑢)) ∘𝑓 +
((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd
‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))} |
101 | | elmapi 8162 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))) →
𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁))) |
102 | 101 | ad3antlr 721 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁))) |
103 | | simpr 479 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) |
104 | | simpl 476 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃𝑝 ∈
ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
105 | 104 | ralimi 3134 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
106 | 105 | ad2antlr 717 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
107 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑚 → (𝑝‘𝑛) = (𝑝‘𝑚)) |
108 | 107 | neeq1d 3028 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑚) ≠ 0)) |
109 | 108 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 0)) |
110 | | fveq1 6445 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = 𝑞 → (𝑝‘𝑚) = (𝑞‘𝑚)) |
111 | 110 | neeq1d 3028 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 0 ↔ (𝑞‘𝑚) ≠ 0)) |
112 | 111 | cbvrexv 3368 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑝 ∈ ran
𝑥(𝑝‘𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) |
113 | 109, 112 | syl6bb 279 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0)) |
114 | 113 | rspccva 3510 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑛 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) |
115 | 106, 114 | sylan 575 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) |
116 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃𝑝 ∈
ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
117 | 116 | ralimi 3134 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
118 | 117 | ad2antlr 717 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
119 | 107 | neeq1d 3028 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 𝐾 ↔ (𝑝‘𝑚) ≠ 𝐾)) |
120 | 119 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 𝐾)) |
121 | 110 | neeq1d 3028 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 𝐾 ↔ (𝑞‘𝑚) ≠ 𝐾)) |
122 | 121 | cbvrexv 3368 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑝 ∈ ran
𝑥(𝑝‘𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) |
123 | 120, 122 | syl6bb 279 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾)) |
124 | 123 | rspccva 3510 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑛 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) |
125 | 118, 124 | sylan 575 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) |
126 | 65, 100, 102, 103, 115, 125 | poimirlem22 34057 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠) |
127 | | eldifsn 4550 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) |
128 | 127 | eubii 2605 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) |
129 | 53 | elexi 3415 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V |
130 | | euhash1 13522 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V →
((♯‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}))) |
131 | 129, 130 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢
((♯‘({𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) |
132 | | df-reu 3097 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃!𝑧 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) |
133 | 128, 131,
132 | 3bitr4ri 296 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃!𝑧 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ (♯‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1) |
134 | 126, 133 | sylib 210 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
(♯‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1) |
135 | 134 | oveq1d 6937 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
((♯‘({𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (1 + (♯‘{𝑠}))) |
136 | 63, 135 | eqtr3d 2816 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
(♯‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (1 +
(♯‘{𝑠}))) |
137 | 50, 136 | syl5breqr 4924 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
138 | 137 | ex 403 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
139 | 138 | exlimdv 1976 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
140 | 41, 139 | syl5bi 234 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (¬ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2
∥ (♯‘{𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
141 | | dvds0 15404 |
. . . . . . . . . . . . . 14
⊢ (2 ∈
ℤ → 2 ∥ 0) |
142 | 9, 141 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ 2 ∥
0 |
143 | | hash0 13473 |
. . . . . . . . . . . . 13
⊢
(♯‘∅) = 0 |
144 | 142, 143 | breqtrri 4913 |
. . . . . . . . . . . 12
⊢ 2 ∥
(♯‘∅) |
145 | | fveq2 6446 |
. . . . . . . . . . . 12
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ →
(♯‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) =
(♯‘∅)) |
146 | 144, 145 | syl5breqr 4924 |
. . . . . . . . . . 11
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2
∥ (♯‘{𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
147 | 140, 146 | pm2.61d2 174 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) |
148 | 147 | ex 403 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
149 | 148 | adantld 486 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) |
150 | | iba 523 |
. . . . . . . . . . 11
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) |
151 | 150 | rabbidv 3386 |
. . . . . . . . . 10
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
152 | 151 | fveq2d 6450 |
. . . . . . . . 9
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
153 | 152 | breq2d 4898 |
. . . . . . . 8
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ↔ 2 ∥
(♯‘{𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) |
154 | 149, 153 | mpbidi 233 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) |
155 | 154 | a1d 25 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})))) |
156 | 34, 40, 155 | rexlimd 3208 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (∃𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) |
157 | 33, 156 | syl5bir 235 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (¬ ∀𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) |
158 | | simpr 479 |
. . . . . . . . 9
⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
159 | 158 | con3i 152 |
. . . . . . . 8
⊢ (¬
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) |
160 | 159 | ralimi 3134 |
. . . . . . 7
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) |
161 | | rabeq0 4187 |
. . . . . . 7
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅ ↔ ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) |
162 | 160, 161 | sylibr 226 |
. . . . . 6
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅) |
163 | 162 | fveq2d 6450 |
. . . . 5
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) =
(♯‘∅)) |
164 | 144, 163 | syl5breqr 4924 |
. . . 4
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
165 | 157, 164 | pm2.61d2 174 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
166 | 8, 10, 32, 165 | fsumdvds 15437 |
. 2
⊢ (𝜑 → 2 ∥ Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))(♯‘{𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
167 | | rabfi 8473 |
. . . . 5
⊢
(((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin) |
168 | 27, 167 | ax-mp 5 |
. . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin |
169 | | simp1 1127 |
. . . . . . 7
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) |
170 | | sneq 4408 |
. . . . . . . . . . . . 13
⊢
((2nd ‘𝑡) = 𝑁 → {(2nd ‘𝑡)} = {𝑁}) |
171 | 170 | difeq2d 3951 |
. . . . . . . . . . . 12
⊢
((2nd ‘𝑡) = 𝑁 → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = ((0...𝑁) ∖ {𝑁})) |
172 | | difun2 4272 |
. . . . . . . . . . . . 13
⊢
(((0...(𝑁 −
1)) ∪ {𝑁}) ∖
{𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁}) |
173 | 64 | nnnn0d 11702 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
174 | | nn0uz 12028 |
. . . . . . . . . . . . . . . . . 18
⊢
ℕ0 = (ℤ≥‘0) |
175 | 173, 174 | syl6eleq 2869 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
176 | | fzm1 12738 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
177 | 175, 176 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
178 | | elun 3976 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁})) |
179 | | velsn 4414 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁) |
180 | 179 | orbi2i 899 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) |
181 | 178, 180 | bitri 267 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) |
182 | 177, 181 | syl6bbr 281 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}))) |
183 | 182 | eqrdv 2776 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁})) |
184 | 183 | difeq1d 3950 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁})) |
185 | 64 | nnzd 11833 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℤ) |
186 | | uzid 12007 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) |
187 | | uznfz 12741 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1))) |
188 | 185, 186,
187 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1))) |
189 | | disjsn 4478 |
. . . . . . . . . . . . . . 15
⊢
(((0...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ ¬ 𝑁 ∈
(0...(𝑁 −
1))) |
190 | | disj3 4246 |
. . . . . . . . . . . . . . 15
⊢
(((0...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ (0...(𝑁 − 1))
= ((0...(𝑁 − 1))
∖ {𝑁})) |
191 | 189, 190 | bitr3i 269 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁})) |
192 | 188, 191 | sylib 210 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁})) |
193 | 172, 184,
192 | 3eqtr4a 2840 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1))) |
194 | 171, 193 | sylan9eqr 2836 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = (0...(𝑁 − 1))) |
195 | 194 | rexeqdv 3341 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
196 | 195 | biimprd 240 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
197 | 196 | ralimdv 3145 |
. . . . . . . 8
⊢ ((𝜑 ∧ (2nd
‘𝑡) = 𝑁) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
198 | 197 | expimpd 447 |
. . . . . . 7
⊢ (𝜑 → (((2nd
‘𝑡) = 𝑁 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
199 | 169, 198 | sylan2i 599 |
. . . . . 6
⊢ (𝜑 → (((2nd
‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
200 | 199 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
201 | 200 | ss2rabdv 3904 |
. . . 4
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) |
202 | | hashssdif 13514 |
. . . 4
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) |
203 | 168, 201,
202 | sylancr 581 |
. . 3
⊢ (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) |
204 | 64 | adantr 474 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → 𝑁 ∈ ℕ) |
205 | | poimirlem28.1 |
. . . . . . . . . 10
⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) |
206 | | poimirlem28.2 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) |
207 | 206 | adantlr 705 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) |
208 | | xp1st 7477 |
. . . . . . . . . . . 12
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
209 | | xp1st 7477 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st
‘(1st ‘𝑡)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁))) |
210 | | elmapi 8162 |
. . . . . . . . . . . 12
⊢
((1st ‘(1st ‘𝑡)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) |
211 | 208, 209,
210 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) |
212 | 211 | adantl 475 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st
‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) |
213 | | xp2nd 7478 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
214 | | fvex 6459 |
. . . . . . . . . . . . . 14
⊢
(2nd ‘(1st ‘𝑡)) ∈ V |
215 | | f1oeq1 6380 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = (2nd
‘(1st ‘𝑡)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))) |
216 | 214, 215 | elab 3558 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(1st ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) |
217 | 213, 216 | sylib 210 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) |
218 | 208, 217 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) |
219 | 218 | adantl 475 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd
‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)) |
220 | | xp2nd 7478 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑡) ∈ (0...𝑁)) |
221 | 220 | adantl 475 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd ‘𝑡) ∈ (0...𝑁)) |
222 | 204, 205,
207, 212, 219, 221 | poimirlem24 34059 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
223 | 208 | adantl 475 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
224 | | 1st2nd2 7484 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑡) = 〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉) |
225 | 224 | csbeq1d 3758 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶) |
226 | 225 | eqeq2d 2788 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) |
227 | 226 | rexbidv 3237 |
. . . . . . . . . . . 12
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) |
228 | 227 | ralbidv 3168 |
. . . . . . . . . . 11
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶)) |
229 | 228 | anbi1d 623 |
. . . . . . . . . 10
⊢
((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
230 | 223, 229 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋〈(1st
‘(1st ‘𝑡)), (2nd ‘(1st
‘𝑡))〉 / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
231 | 222, 230 | bitr4d 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
232 | 101 | frnd 6298 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))) →
ran 𝑥 ⊆ ((0...𝐾) ↑𝑚
(1...𝑁))) |
233 | 232 | anim2i 610 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)))) |
234 | | dfss3 3810 |
. . . . . . . . . . . . . 14
⊢
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) |
235 | | vex 3401 |
. . . . . . . . . . . . . . . 16
⊢ 𝑛 ∈ V |
236 | | eqid 2778 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ ran 𝑥 ↦ 𝐵) = (𝑝 ∈ ran 𝑥 ↦ 𝐵) |
237 | 236 | elrnmpt 5618 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
238 | 235, 237 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) |
239 | 238 | ralbii 3162 |
. . . . . . . . . . . . . 14
⊢
(∀𝑛 ∈
(0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) |
240 | 234, 239 | sylbb 211 |
. . . . . . . . . . . . 13
⊢
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) |
241 | | 1eluzge0 12038 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
(ℤ≥‘0) |
242 | | fzss1 12697 |
. . . . . . . . . . . . . . . . 17
⊢ (1 ∈
(ℤ≥‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))) |
243 | | ssralv 3885 |
. . . . . . . . . . . . . . . . 17
⊢
((1...(𝑁 − 1))
⊆ (0...(𝑁 − 1))
→ (∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
244 | 241, 242,
243 | mp2b 10 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) |
245 | 64 | nncnd 11392 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 ∈ ℂ) |
246 | | npcan1 10800 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
247 | 245, 246 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
248 | | peano2zm 11772 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
249 | 185, 248 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
250 | | uzid 12007 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) |
251 | | peano2uz 12047 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
252 | 249, 250,
251 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
253 | 247, 252 | eqeltrrd 2860 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
254 | | fzss2 12698 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈
(ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
255 | 253, 254 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
256 | 255 | sselda 3821 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁)) |
257 | 256 | adantlr 705 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁)) |
258 | | simplr 759 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) |
259 | | ssel2 3816 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ran
𝑥 ⊆ ((0...𝐾) ↑𝑚
(1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑𝑚 (1...𝑁))) |
260 | | elmapi 8162 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 ∈ ((0...𝐾) ↑𝑚 (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾)) |
261 | 259, 260 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ran
𝑥 ⊆ ((0...𝐾) ↑𝑚
(1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾)) |
262 | 258, 261 | sylan 575 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾)) |
263 | | poimirlem28.3 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) |
264 | | elfzelz 12659 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ) |
265 | 264 | zred 11834 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ) |
266 | 265 | ltnrd 10510 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛) |
267 | | breq1 4889 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 = 𝐵 → (𝑛 < 𝑛 ↔ 𝐵 < 𝑛)) |
268 | 267 | notbid 310 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛)) |
269 | 266, 268 | syl5ibcom 237 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛)) |
270 | 269 | necon2ad 2984 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) |
271 | 270 | 3ad2ant1 1124 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) |
272 | 271 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) |
273 | 263, 272 | mpd 15 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝑛 ≠ 𝐵) |
274 | 273 | 3exp2 1416 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵)))) |
275 | 274 | imp31 410 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵)) |
276 | 275 | necon2d 2992 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) |
277 | 276 | adantllr 709 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) |
278 | 262, 277 | syldan 585 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) |
279 | 278 | reximdva 3198 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
280 | 257, 279 | syldan 585 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
281 | 280 | ralimdva 3144 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) → (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
282 | 281 | imp 397 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
283 | 244, 282 | sylan2 586 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) |
284 | 283 | biantrurd 528 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) |
285 | | nnuz 12029 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ =
(ℤ≥‘1) |
286 | 64, 285 | syl6eleq 2869 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
287 | | fzm1 12738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
288 | 286, 287 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) |
289 | | elun 3976 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁})) |
290 | 179 | orbi2i 899 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) |
291 | 289, 290 | bitri 267 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) |
292 | 288, 291 | syl6bbr 281 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}))) |
293 | 292 | eqrdv 2776 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
294 | 293 | raleqdv 3340 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
295 | | ralunb 4017 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑛 ∈
((1...(𝑁 − 1)) ∪
{𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) |
296 | 294, 295 | syl6bb 279 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))) |
297 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑁 → (𝑝‘𝑛) = (𝑝‘𝑁)) |
298 | 297 | neeq1d 3028 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 𝑁 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑁) ≠ 0)) |
299 | 298 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) |
300 | 299 | ralsng 4444 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ →
(∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) |
301 | 64, 300 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) |
302 | 301 | anbi2d 622 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) |
303 | 296, 302 | bitrd 271 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) |
304 | 303 | ad2antrr 716 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) |
305 | | 0z 11739 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 ∈
ℤ |
306 | | 1z 11759 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℤ |
307 | | fzshftral 12746 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((0
∈ ℤ ∧ (𝑁
− 1) ∈ ℤ ∧ 1 ∈ ℤ) → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
308 | 305, 306,
307 | mp3an13 1525 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑁 − 1) ∈ ℤ
→ (∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
309 | 185, 248,
308 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
310 | | 0p1e1 11504 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 + 1) =
1 |
311 | 310 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (0 + 1) =
1) |
312 | 311, 247 | oveq12d 6940 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁)) |
313 | 312 | raleqdv 3340 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
314 | 309, 313 | bitrd 271 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) |
315 | | ovex 6954 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 − 1) ∈
V |
316 | | eqeq1 2782 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵)) |
317 | 316 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)) |
318 | 315, 317 | sbcie 3687 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
([(𝑚 −
1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵) |
319 | 318 | ralbii 3162 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
(1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵) |
320 | | oveq1 6929 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1)) |
321 | 320 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵)) |
322 | 321 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)) |
323 | 322 | cbvralv 3367 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) |
324 | 319, 323 | bitri 267 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑚 ∈
(1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) |
325 | 314, 324 | syl6bb 279 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)) |
326 | 325 | biimpa 470 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) |
327 | 326 | adantlr 705 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) |
328 | | poimirlem28.4 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) |
329 | 328 | necomd 3024 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵) |
330 | 329 | 3exp2 1416 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵)))) |
331 | 330 | imp31 410 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵)) |
332 | 331 | necon2d 2992 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) |
333 | 332 | adantllr 709 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) |
334 | 262, 333 | syldan 585 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) |
335 | 334 | reximdva 3198 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) |
336 | 335 | ralimdva 3144 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) |
337 | 336 | imp 397 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
338 | 327, 337 | syldan 585 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) |
339 | 338 | biantrud 527 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
340 | | r19.26 3250 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) |
341 | 339, 340 | syl6bbr 281 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
342 | 284, 304,
341 | 3bitr2d 299 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
343 | 233, 240,
342 | syl2an 589 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))) ∧
(0...(𝑁 − 1)) ⊆
ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) |
344 | 343 | pm5.32da 574 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) |
345 | 344 | anbi2d 622 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) |
346 | 345 | rexbidva 3234 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) |
347 | 346 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) |
348 | 193 | rexeqdv 3341 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
349 | 348 | biimpd 221 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
350 | 349 | ralimdv 3145 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
351 | 171 | rexeqdv 3341 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
352 | 351 | ralbidv 3168 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
353 | 352 | imbi1d 333 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘𝑡) = 𝑁 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
354 | 350, 353 | syl5ibrcom 239 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
355 | 354 | com23 86 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 → ((2nd ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶))) |
356 | 355 | imp 397 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → ((2nd ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
357 | 356 | adantrd 487 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶)) |
358 | 357 | pm4.71rd 558 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
359 | | an12 635 |
. . . . . . . . . . . . 13
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) |
360 | | 3anass 1079 |
. . . . . . . . . . . . . 14
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) |
361 | 360 | anbi2i 616 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) |
362 | 359, 361 | bitr4i 270 |
. . . . . . . . . . . 12
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) |
363 | 358, 362 | syl6bb 279 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) |
364 | 363 | notbid 310 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶) → (¬ ((2nd
‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ¬ ((2nd
‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))) |
365 | 364 | pm5.32da 574 |
. . . . . . . . 9
⊢ (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
366 | 365 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
367 | 231, 347,
366 | 3bitr3d 301 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))))) |
368 | 367 | rabbidva 3385 |
. . . . . 6
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))}) |
369 | | iunrab 4800 |
. . . . . 6
⊢ ∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} |
370 | | difrab 4127 |
. . . . . 6
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)))} |
371 | 368, 369,
370 | 3eqtr4g 2839 |
. . . . 5
⊢ (𝜑 → ∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) |
372 | 371 | fveq2d 6450 |
. . . 4
⊢ (𝜑 → (♯‘∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = (♯‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}))) |
373 | 27, 28 | mp1i 13 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))))
→ {𝑡 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin) |
374 | | simpl 476 |
. . . . . . . . . . . 12
⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
375 | 374 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
376 | 375 | ss2rabi 3905 |
. . . . . . . . . 10
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
377 | 376 | sseli 3817 |
. . . . . . . . 9
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) |
378 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (2nd ‘𝑡) = (2nd ‘𝑠)) |
379 | 378 | breq2d 4898 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑠))) |
380 | 379 | ifbid 4329 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑠 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1))) |
381 | 380 | csbeq1d 3758 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
382 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑠))) |
383 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑠 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑠))) |
384 | 383 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑠 → ((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st
‘𝑠)) “
(1...𝑗))) |
385 | 384 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1})) |
386 | 383 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑠 → ((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st
‘𝑠)) “ ((𝑗 + 1)...𝑁))) |
387 | 386 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})) |
388 | 385, 387 | uneq12d 3991 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → ((((2nd
‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd
‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
389 | 382, 388 | oveq12d 6940 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑠 → ((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
390 | 389 | csbeq2dv 4217 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
391 | 381, 390 | eqtrd 2814 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
392 | 391 | mpteq2dv 4980 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑠 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
393 | 392 | eqeq2d 2788 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
394 | | eqcom 2785 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) |
395 | 393, 394 | syl6bb 279 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)) |
396 | 395 | elrab 3572 |
. . . . . . . . . 10
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ↔ (𝑠 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)) |
397 | 396 | simprbi 492 |
. . . . . . . . 9
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) |
398 | 377, 397 | syl 17 |
. . . . . . . 8
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) |
399 | 398 | rgen 3104 |
. . . . . . 7
⊢
∀𝑠 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 |
400 | 399 | rgenw 3106 |
. . . . . 6
⊢
∀𝑥 ∈
(((0...𝐾)
↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 |
401 | | invdisj 4872 |
. . . . . 6
⊢
(∀𝑥 ∈
(((0...𝐾)
↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑠)) ∘𝑓 +
((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 → Disj 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
402 | 400, 401 | mp1i 13 |
. . . . 5
⊢ (𝜑 → Disj 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) |
403 | 8, 373, 402 | hashiun 14958 |
. . . 4
⊢ (𝜑 → (♯‘∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))(♯‘{𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
404 | 372, 403 | eqtr3d 2816 |
. . 3
⊢ (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚
(0...(𝑁 −
1)))(♯‘{𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st
‘(1st ‘𝑡)) ∘𝑓 +
((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd
‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) |
405 | | fo1st 7465 |
. . . . . . . . . . . . 13
⊢
1st :V–onto→V |
406 | | fofun 6367 |
. . . . . . . . . . . . 13
⊢
(1st :V–onto→V → Fun 1st ) |
407 | 405, 406 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ Fun
1st |
408 | | ssv 3844 |
. . . . . . . . . . . . 13
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ V |
409 | | fof 6366 |
. . . . . . . . . . . . . . 15
⊢
(1st :V–onto→V → 1st
:V⟶V) |
410 | 405, 409 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢
1st :V⟶V |
411 | 410 | fdmi 6301 |
. . . . . . . . . . . . 13
⊢ dom
1st = V |
412 | 408, 411 | sseqtr4i 3857 |
. . . . . . . . . . . 12
⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom
1st |
413 | | fores 6376 |
. . . . . . . . . . . 12
⊢ ((Fun
1st ∧ {𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) →
(1st ↾ {𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))})) |
414 | 407, 412,
413 | mp2an 682 |
. . . . . . . . . . 11
⊢
(1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) |
415 | | fveqeq2 6455 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ((2nd ‘𝑡) = 𝑁 ↔ (2nd ‘𝑥) = 𝑁)) |
416 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 = 𝑥 → (1st ‘𝑡) = (1st ‘𝑥)) |
417 | 416 | csbeq1d 3758 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑥 → ⦋(1st
‘𝑡) / 𝑠⦌𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
418 | 417 | eqeq2d 2788 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
419 | 418 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
420 | 419 | ralbidv 3168 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶)) |
421 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (1st
‘(1st ‘𝑡)) = (1st ‘(1st
‘𝑥))) |
422 | 421 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ((1st
‘(1st ‘𝑡))‘𝑁) = ((1st ‘(1st
‘𝑥))‘𝑁)) |
423 | 422 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (((1st
‘(1st ‘𝑡))‘𝑁) = 0 ↔ ((1st
‘(1st ‘𝑥))‘𝑁) = 0)) |
424 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (2nd
‘(1st ‘𝑡)) = (2nd ‘(1st
‘𝑥))) |
425 | 424 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ((2nd
‘(1st ‘𝑡))‘𝑁) = ((2nd ‘(1st
‘𝑥))‘𝑁)) |
426 | 425 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁 ↔ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) |
427 | 420, 423,
426 | 3anbi123d 1509 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) |
428 | 415, 427 | anbi12d 624 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑥 → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)))) |
429 | 428 | rexrab 3580 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑥 ∈
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) |
430 | | xp1st 7477 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
431 | 430 | anim1i 608 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) |
432 | | eleq1 2847 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥) = 𝑠 → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))) |
433 | | csbeq1a 3760 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 = (1st ‘𝑥) → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
434 | 433 | eqcoms 2786 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((1st ‘𝑥) = 𝑠 → 𝐶 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶) |
435 | 434 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((1st ‘𝑥) = 𝑠 → ⦋(1st
‘𝑥) / 𝑠⦌𝐶 = 𝐶) |
436 | 435 | eqeq2d 2788 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) |
437 | 436 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) |
438 | 437 | ralbidv 3168 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) |
439 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (1st
‘(1st ‘𝑥)) = (1st ‘𝑠)) |
440 | 439 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → ((1st
‘(1st ‘𝑥))‘𝑁) = ((1st ‘𝑠)‘𝑁)) |
441 | 440 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (((1st
‘(1st ‘𝑥))‘𝑁) = 0 ↔ ((1st ‘𝑠)‘𝑁) = 0)) |
442 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑥) = 𝑠 → (2nd
‘(1st ‘𝑥)) = (2nd ‘𝑠)) |
443 | 442 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑥) = 𝑠 → ((2nd
‘(1st ‘𝑥))‘𝑁) = ((2nd ‘𝑠)‘𝑁)) |
444 | 443 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1st ‘𝑥) = 𝑠 → (((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘𝑠)‘𝑁) = 𝑁)) |
445 | 438, 441,
444 | 3anbi123d 1509 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥) = 𝑠 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) |
446 | 432, 445 | anbi12d 624 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥) = 𝑠 → (((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
447 | 431, 446 | syl5ibcom 237 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
448 | 447 | adantrl 706 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁))) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
449 | 448 | expimpd 447 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
450 | 449 | rexlimiv 3209 |
. . . . . . . . . . . . . . . 16
⊢
(∃𝑥 ∈
((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) |
451 | | nn0fz0 12756 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ0
↔ 𝑁 ∈ (0...𝑁)) |
452 | 173, 451 | sylib 210 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
453 | | opelxpi 5392 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
454 | 452, 453 | sylan2 586 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝜑) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
455 | 454 | ancoms 452 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
456 | | opelxp2 5397 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁)) |
457 | | op2ndg 7458 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘〈𝑠, 𝑁〉) = 𝑁) |
458 | 457 | biantrurd 528 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁) ↔ ((2nd ‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)))) |
459 | | op1stg 7457 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘〈𝑠, 𝑁〉) = 𝑠) |
460 | | csbeq1a 3760 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑠 = (1st
‘〈𝑠, 𝑁〉) → 𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) |
461 | 460 | eqcoms 2786 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((1st ‘〈𝑠, 𝑁〉) = 𝑠 → 𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) |
462 | 461 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((1st ‘〈𝑠, 𝑁〉) = 𝑠 → ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 = 𝐶) |
463 | 459, 462 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 = 𝐶) |
464 | 463 | eqeq2d 2788 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) |
465 | 464 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) |
466 | 465 | ralbidv 3168 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) |
467 | 459 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st
‘(1st ‘〈𝑠, 𝑁〉)) = (1st ‘𝑠)) |
468 | 467 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = ((1st ‘𝑠)‘𝑁)) |
469 | 468 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ↔ ((1st ‘𝑠)‘𝑁) = 0)) |
470 | 459 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd
‘(1st ‘〈𝑠, 𝑁〉)) = (2nd ‘𝑠)) |
471 | 470 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = ((2nd ‘𝑠)‘𝑁)) |
472 | 471 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁 ↔ ((2nd ‘𝑠)‘𝑁) = 𝑁)) |
473 | 466, 469,
472 | 3anbi123d 1509 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))) |
474 | 459 | biantrud 527 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) |
475 | 458, 473,
474 | 3bitr3d 301 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) |
476 | 44, 456, 475 | sylancr 581 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁) ↔ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) |
477 | 476 | biimpa 470 |
. . . . . . . . . . . . . . . . . . 19
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) |
478 | | fveqeq2 6455 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((2nd ‘𝑥) = 𝑁 ↔ (2nd ‘〈𝑠, 𝑁〉) = 𝑁)) |
479 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (1st ‘𝑥) = (1st
‘〈𝑠, 𝑁〉)) |
480 | 479 | csbeq1d 3758 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ⦋(1st
‘𝑥) / 𝑠⦌𝐶 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶) |
481 | 480 | eqeq2d 2788 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) |
482 | 481 | rexbidv 3237 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) |
483 | 482 | ralbidv 3168 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶)) |
484 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (1st
‘(1st ‘𝑥)) = (1st ‘(1st
‘〈𝑠, 𝑁〉))) |
485 | 484 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((1st
‘(1st ‘𝑥))‘𝑁) = ((1st ‘(1st
‘〈𝑠, 𝑁〉))‘𝑁)) |
486 | 485 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((1st
‘(1st ‘𝑥))‘𝑁) = 0 ↔ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0)) |
487 | | 2fveq3 6451 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (2nd
‘(1st ‘𝑥)) = (2nd ‘(1st
‘〈𝑠, 𝑁〉))) |
488 | 487 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((2nd
‘(1st ‘𝑥))‘𝑁) = ((2nd ‘(1st
‘〈𝑠, 𝑁〉))‘𝑁)) |
489 | 488 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁 ↔ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) |
490 | 483, 486,
489 | 3anbi123d 1509 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁))) |
491 | 478, 490 | anbi12d 624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑠, 𝑁〉 → (((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ↔ ((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)))) |
492 | | fveqeq2 6455 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((1st ‘𝑥) = 𝑠 ↔ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) |
493 | 491, 492 | anbi12d 624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 〈𝑠, 𝑁〉 → ((((2nd
‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) ↔ (((2nd ‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠))) |
494 | 493 | rspcev 3511 |
. . . . . . . . . . . . . . . . . . 19
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (((2nd
‘〈𝑠, 𝑁〉) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘〈𝑠, 𝑁〉) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘〈𝑠, 𝑁〉))‘𝑁) = 𝑁)) ∧ (1st ‘〈𝑠, 𝑁〉) = 𝑠)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) |
495 | 477, 494 | syldan 585 |
. . . . . . . . . . . . . . . . . 18
⊢
((〈𝑠, 𝑁〉 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) |
496 | 455, 495 | sylan 575 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)) |
497 | 496 | expl 451 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠))) |
498 | 450, 497 | impbid2 218 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑥) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
499 | 429, 498 | syl5bb 275 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))) |
500 | 499 | abbidv 2906 |
. . . . . . . . . . . . 13
⊢ (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))}) |
501 | | dfimafn 6505 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
1st ∧ {𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) →
(1st “ {𝑡
∈ ((((0..^𝐾)
↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦}) |
502 | 407, 412,
501 | mp2an 682 |
. . . . . . . . . . . . . 14
⊢
(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦} |
503 | | nfv 1957 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(2nd ‘𝑡) = 𝑁 |
504 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠(0...(𝑁 − 1)) |
505 | | nfcsb1v 3767 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑠⦋(1st ‘𝑡) / 𝑠⦌𝐶 |
506 | 505 | nfeq2 2949 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑠 𝑖 =
⦋(1st ‘𝑡) / 𝑠⦌𝐶 |
507 | 504, 506 | nfrex 3188 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
508 | 504, 507 | nfral 3127 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 |
509 | | nfv 1957 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠((1st ‘(1st
‘𝑡))‘𝑁) = 0 |
510 | | nfv 1957 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠((2nd ‘(1st
‘𝑡))‘𝑁) = 𝑁 |
511 | 508, 509,
510 | nf3an 1948 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁) |
512 | 503, 511 | nfan 1946 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁)) |
513 | | nfcv 2934 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) |
514 | 512, 513 | nfrab 3310 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} |
515 | | nfv 1957 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠(1st ‘𝑥) = 𝑦 |
516 | 514, 515 | nfrex 3188 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑠∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦 |
517 | | nfv 1957 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑦∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 |
518 | | eqeq2 2789 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑠 → ((1st ‘𝑥) = 𝑦 ↔ (1st ‘𝑥) = 𝑠)) |
519 | 518 | rexbidv 3237 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠)) |
520 | 516, 517,
519 | cbvab 2913 |
. . . . . . . . . . . . . 14
⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} |
521 | 502, 520 | eqtri 2802 |
. . . . . . . . . . . . 13
⊢
(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} |
522 | | df-rab 3099 |
. . . . . . . . . . . . 13
⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))} |
523 | 500, 521,
522 | 3eqtr4g 2839 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1st “
{𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) |
524 | | foeq3 6364 |
. . . . . . . . . . . 12
⊢
((1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌𝐶 ∧ ((1st
‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd
‘(1st ‘𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌ |