Step | Hyp | Ref
| Expression |
1 | | poimir.0 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℕ) |
2 | 1 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑁 ∈ ℕ) |
3 | | poimirlem4.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ ℕ) |
4 | 3 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝐾 ∈ ℕ) |
5 | | poimirlem4.2 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
6 | 5 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 ∈
ℕ0) |
7 | | poimirlem4.3 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 < 𝑁) |
8 | 7 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 < 𝑁) |
9 | | xp1st 7477 |
. . . . . . . . 9
⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st ‘𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀))) |
10 | | elmapi 8162 |
. . . . . . . . 9
⊢
((1st ‘𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st
‘𝑡):(1...𝑀)⟶(0..^𝐾)) |
11 | 9, 10 | syl 17 |
. . . . . . . 8
⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st ‘𝑡):(1...𝑀)⟶(0..^𝐾)) |
12 | 11 | adantl 475 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (1st ‘𝑡):(1...𝑀)⟶(0..^𝐾)) |
13 | | xp2nd 7478 |
. . . . . . . . 9
⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd ‘𝑡) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) |
14 | | fvex 6459 |
. . . . . . . . . 10
⊢
(2nd ‘𝑡) ∈ V |
15 | | f1oeq1 6380 |
. . . . . . . . . 10
⊢ (𝑓 = (2nd ‘𝑡) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀))) |
16 | 14, 15 | elab 3558 |
. . . . . . . . 9
⊢
((2nd ‘𝑡) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀)) |
17 | 13, 16 | sylib 210 |
. . . . . . . 8
⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀)) |
18 | 17 | adantl 475 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀)) |
19 | 2, 4, 6, 8, 12, 18 | poimirlem3 34038 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
∈ (((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉})‘(𝑀 + 1)) = 0 ∧
(((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})‘(𝑀 + 1)) =
(𝑀 +
1))))) |
20 | | fvex 6459 |
. . . . . . . . . . . . . . . 16
⊢
(1st ‘𝑡) ∈ V |
21 | | snex 5140 |
. . . . . . . . . . . . . . . 16
⊢
{〈(𝑀 + 1),
0〉} ∈ V |
22 | 20, 21 | unex 7233 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) ∈ V |
23 | | snex 5140 |
. . . . . . . . . . . . . . . 16
⊢
{〈(𝑀 + 1),
(𝑀 + 1)〉} ∈
V |
24 | 14, 23 | unex 7233 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) ∈ V |
25 | 22, 24 | op1std 7455 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (1st ‘𝑠) = ((1st ‘𝑡) ∪ {〈(𝑀 + 1),
0〉})) |
26 | 22, 24 | op2ndd 7456 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (2nd ‘𝑠) = ((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉})) |
27 | 26 | imaeq1d 5719 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ ((2nd ‘𝑠) “ (1...𝑗)) = (((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗))) |
28 | 27 | xpeq1d 5384 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = ((((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) ×
{1})) |
29 | 26 | imaeq1d 5719 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1)))) |
30 | 29 | xpeq1d 5384 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) |
31 | 28, 30 | uneq12d 3991 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) |
32 | 25, 31 | oveq12d 6940 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})))) |
33 | 32 | uneq1d 3989 |
. . . . . . . . . . . 12
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))) |
34 | 33 | csbeq1d 3758 |
. . . . . . . . . . 11
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ ⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
35 | 34 | eqeq2d 2788 |
. . . . . . . . . 10
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (𝑖 =
⦋(((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋((((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
36 | 35 | rexbidv 3237 |
. . . . . . . . 9
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (∃𝑗 ∈
(0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
37 | 36 | ralbidv 3168 |
. . . . . . . 8
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (∀𝑖 ∈
(0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
38 | 25 | fveq1d 6448 |
. . . . . . . . 9
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ ((1st ‘𝑠)‘(𝑀 + 1)) = (((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉})‘(𝑀 + 1))) |
39 | 38 | eqeq1d 2780 |
. . . . . . . 8
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (((1st ‘𝑠)‘(𝑀 + 1)) = 0 ↔ (((1st
‘𝑡) ∪
{〈(𝑀 + 1),
0〉})‘(𝑀 + 1)) =
0)) |
40 | 26 | fveq1d 6448 |
. . . . . . . . 9
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ ((2nd ‘𝑠)‘(𝑀 + 1)) = (((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉})‘(𝑀 + 1))) |
41 | 40 | eqeq1d 2780 |
. . . . . . . 8
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ (((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉})‘(𝑀 + 1)) = (𝑀 + 1))) |
42 | 37, 39, 41 | 3anbi123d 1509 |
. . . . . . 7
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ ((∀𝑖 ∈
(0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉})‘(𝑀 + 1)) = 0 ∧
(((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})‘(𝑀 + 1)) =
(𝑀 + 1)))) |
43 | 42 | elrab 3572 |
. . . . . 6
⊢
(〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})〉 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...(𝑀 + 1))) ×
{𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ (〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
∈ (((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉})
∘𝑓 + (((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉})‘(𝑀 + 1)) = 0 ∧
(((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})‘(𝑀 + 1)) =
(𝑀 + 1)))) |
44 | 19, 43 | syl6ibr 244 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
∈ {𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})) |
45 | 44 | ralrimiva 3148 |
. . . 4
⊢ (𝜑 → ∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
∈ {𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})) |
46 | | fveq2 6446 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑡 → (1st ‘𝑠) = (1st ‘𝑡)) |
47 | | fveq2 6446 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑡 → (2nd ‘𝑠) = (2nd ‘𝑡)) |
48 | 47 | imaeq1d 5719 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd
‘𝑡) “
(1...𝑗))) |
49 | 48 | xpeq1d 5384 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) =
(((2nd ‘𝑡)
“ (1...𝑗)) ×
{1})) |
50 | 47 | imaeq1d 5719 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀))) |
51 | 50 | xpeq1d 5384 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0})) |
52 | 49, 51 | uneq12d 3991 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑡 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑀)) × {0})) =
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) |
53 | 46, 52 | oveq12d 6940 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑡 → ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) = ((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0})))) |
54 | 53 | uneq1d 3989 |
. . . . . . . . 9
⊢ (𝑠 = 𝑡 → (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0}))) |
55 | 54 | csbeq1d 3758 |
. . . . . . . 8
⊢ (𝑠 = 𝑡 → ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
56 | 55 | eqeq2d 2788 |
. . . . . . 7
⊢ (𝑠 = 𝑡 → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
57 | 56 | rexbidv 3237 |
. . . . . 6
⊢ (𝑠 = 𝑡 → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
58 | 57 | ralbidv 3168 |
. . . . 5
⊢ (𝑠 = 𝑡 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
59 | 58 | ralrab 3578 |
. . . 4
⊢
(∀𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
∈ {𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ ∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
∈ {𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})) |
60 | 45, 59 | sylibr 226 |
. . 3
⊢ (𝜑 → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
∈ {𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}) |
61 | | xp1st 7477 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st ‘𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1)))) |
62 | | elmapi 8162 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) → (1st
‘𝑘):(1...(𝑀 + 1))⟶(0..^𝐾)) |
63 | 61, 62 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st ‘𝑘):(1...(𝑀 + 1))⟶(0..^𝐾)) |
64 | | fzssp1 12701 |
. . . . . . . . . . . . . 14
⊢
(1...𝑀) ⊆
(1...(𝑀 +
1)) |
65 | | fssres 6320 |
. . . . . . . . . . . . . 14
⊢
(((1st ‘𝑘):(1...(𝑀 + 1))⟶(0..^𝐾) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st ‘𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾)) |
66 | 63, 64, 65 | sylancl 580 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st
‘𝑘) ↾
(1...𝑀)):(1...𝑀)⟶(0..^𝐾)) |
67 | | ovex 6954 |
. . . . . . . . . . . . . 14
⊢
(0..^𝐾) ∈
V |
68 | | ovex 6954 |
. . . . . . . . . . . . . 14
⊢
(1...𝑀) ∈
V |
69 | 67, 68 | elmap 8169 |
. . . . . . . . . . . . 13
⊢
(((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) ↔ ((1st
‘𝑘) ↾
(1...𝑀)):(1...𝑀)⟶(0..^𝐾)) |
70 | 66, 69 | sylibr 226 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st
‘𝑘) ↾
(1...𝑀)) ∈ ((0..^𝐾) ↑𝑚
(1...𝑀))) |
71 | 70 | ad2antlr 717 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀))) |
72 | | xp2nd 7478 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd ‘𝑘) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) |
73 | | fvex 6459 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘𝑘) ∈ V |
74 | | f1oeq1 6380 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 = (2nd ‘𝑘) → (𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)))) |
75 | 73, 74 | elab 3558 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘𝑘) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))) |
76 | 72, 75 | sylib 210 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))) |
77 | | f1of1 6390 |
. . . . . . . . . . . . . . . 16
⊢
((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd ‘𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1))) |
78 | 76, 77 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd ‘𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1))) |
79 | | f1ores 6405 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀))) |
80 | 78, 64, 79 | sylancl 580 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd
‘𝑘) ↾
(1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀))) |
81 | 80 | ad2antlr 717 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀))) |
82 | | dff1o3 6397 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ ((2nd ‘𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)) ∧ Fun ◡(2nd ‘𝑘))) |
83 | 82 | simprbi 492 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → Fun ◡(2nd ‘𝑘)) |
84 | | imadif 6218 |
. . . . . . . . . . . . . . . . . 18
⊢ (Fun
◡(2nd ‘𝑘) → ((2nd
‘𝑘) “
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
(((2nd ‘𝑘)
“ (1...(𝑀 + 1)))
∖ ((2nd ‘𝑘) “ {(𝑀 + 1)}))) |
85 | 76, 83, 84 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd
‘𝑘) “
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
(((2nd ‘𝑘)
“ (1...(𝑀 + 1)))
∖ ((2nd ‘𝑘) “ {(𝑀 + 1)}))) |
86 | 85 | ad2antlr 717 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd
‘𝑘) “
(1...(𝑀 + 1))) ∖
((2nd ‘𝑘)
“ {(𝑀 +
1)}))) |
87 | | f1ofo 6398 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd ‘𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1))) |
88 | | foima 6371 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1))) |
89 | 76, 87, 88 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd
‘𝑘) “
(1...(𝑀 + 1))) =
(1...(𝑀 +
1))) |
90 | 89 | ad2antlr 717 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1))) |
91 | | f1ofn 6392 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd ‘𝑘) Fn (1...(𝑀 + 1))) |
92 | 76, 91 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd ‘𝑘) Fn (1...(𝑀 + 1))) |
93 | | nn0p1nn 11683 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℕ0
→ (𝑀 + 1) ∈
ℕ) |
94 | 5, 93 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 + 1) ∈ ℕ) |
95 | | elfz1end 12688 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 + 1) ∈ ℕ ↔
(𝑀 + 1) ∈ (1...(𝑀 + 1))) |
96 | 94, 95 | sylib 210 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1))) |
97 | | fnsnfv 6518 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2nd ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → {((2nd
‘𝑘)‘(𝑀 + 1))} = ((2nd
‘𝑘) “ {(𝑀 + 1)})) |
98 | 92, 96, 97 | syl2anr 590 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → {((2nd
‘𝑘)‘(𝑀 + 1))} = ((2nd
‘𝑘) “ {(𝑀 + 1)})) |
99 | | sneq 4408 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {((2nd ‘𝑘)‘(𝑀 + 1))} = {(𝑀 + 1)}) |
100 | 98, 99 | sylan9req 2835 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ {(𝑀 + 1)}) = {(𝑀 + 1)}) |
101 | 90, 100 | difeq12d 3952 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd
‘𝑘) “ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) |
102 | 86, 101 | eqtrd 2814 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) |
103 | | 1z 11759 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 1 ∈
ℤ |
104 | | nn0uz 12028 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
ℕ0 = (ℤ≥‘0) |
105 | | 1m1e0 11447 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (1
− 1) = 0 |
106 | 105 | fveq2i 6449 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(ℤ≥‘(1 − 1)) =
(ℤ≥‘0) |
107 | 104, 106 | eqtr4i 2805 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
ℕ0 = (ℤ≥‘(1 −
1)) |
108 | 5, 107 | syl6eleq 2869 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(1
− 1))) |
109 | | fzsuc2 12716 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((1
∈ ℤ ∧ 𝑀
∈ (ℤ≥‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)})) |
110 | 103, 108,
109 | sylancr 581 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)})) |
111 | 110 | difeq1d 3950 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)})) |
112 | | difun2 4272 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...𝑀) ∪
{(𝑀 + 1)}) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)}) |
113 | 111, 112 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)})) |
114 | 5 | nn0red 11703 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑀 ∈ ℝ) |
115 | | ltp1 11215 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀 ∈ ℝ → 𝑀 < (𝑀 + 1)) |
116 | | peano2re 10549 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈
ℝ) |
117 | | ltnle 10456 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ) →
(𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀)) |
118 | 116, 117 | mpdan 677 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀 ∈ ℝ → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀)) |
119 | 115, 118 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℝ → ¬
(𝑀 + 1) ≤ 𝑀) |
120 | 114, 119 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀) |
121 | | elfzle2 12662 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀) |
122 | 120, 121 | nsyl 138 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀)) |
123 | | difsn 4560 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬
(𝑀 + 1) ∈ (1...𝑀) → ((1...𝑀) ∖ {(𝑀 + 1)}) = (1...𝑀)) |
124 | 122, 123 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...𝑀) ∖ {(𝑀 + 1)}) = (1...𝑀)) |
125 | 113, 124 | eqtrd 2814 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀)) |
126 | 125 | imaeq2d 5720 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑘) “
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
((2nd ‘𝑘)
“ (1...𝑀))) |
127 | 126 | ad2antrr 716 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd
‘𝑘) “
(1...𝑀))) |
128 | 125 | ad2antrr 716 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀)) |
129 | 102, 127,
128 | 3eqtr3d 2822 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ (1...𝑀)) = (1...𝑀)) |
130 | | f1oeq3 6382 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘𝑘) “ (1...𝑀)) = (1...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀)) ↔ ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))) |
131 | 129, 130 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀)) ↔ ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))) |
132 | 81, 131 | mpbid 224 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)) |
133 | 73 | resex 5693 |
. . . . . . . . . . . . 13
⊢
((2nd ‘𝑘) ↾ (1...𝑀)) ∈ V |
134 | | f1oeq1 6380 |
. . . . . . . . . . . . 13
⊢ (𝑓 = ((2nd ‘𝑘) ↾ (1...𝑀)) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))) |
135 | 133, 134 | elab 3558 |
. . . . . . . . . . . 12
⊢
(((2nd ‘𝑘) ↾ (1...𝑀)) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)) |
136 | 132, 135 | sylibr 226 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) ↾ (1...𝑀)) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) |
137 | | opelxpi 5392 |
. . . . . . . . . . 11
⊢
((((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) ∧ ((2nd
‘𝑘) ↾
(1...𝑀)) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) |
138 | 71, 136, 137 | syl2anc 579 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) |
139 | 138 | 3ad2antr3 1198 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) |
140 | | 3anass 1079 |
. . . . . . . . . . 11
⊢
((∀𝑖 ∈
(0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)))) |
141 | | ancom 454 |
. . . . . . . . . . 11
⊢
((∀𝑖 ∈
(0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ↔ ((((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
142 | 140, 141 | bitri 267 |
. . . . . . . . . 10
⊢
((∀𝑖 ∈
(0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ ((((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
143 | 94 | nnzd 11833 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
144 | | uzid 12007 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑀 + 1) ∈ ℤ →
(𝑀 + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
145 | | peano2uz 12047 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑀 + 1) ∈
(ℤ≥‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
146 | 143, 144,
145 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((𝑀 + 1) + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
147 | 5 | nn0zd 11832 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑀 ∈ ℤ) |
148 | 1 | nnzd 11833 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑁 ∈ ℤ) |
149 | | zltp1le 11779 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
150 | | peano2z 11770 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈
ℤ) |
151 | | eluz 12006 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁)) |
152 | 150, 151 | sylan 575 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁)) |
153 | 149, 152 | bitr4d 274 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) |
154 | 147, 148,
153 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) |
155 | 7, 154 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) |
156 | | fzsplit2 12683 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 + 1) + 1) ∈
(ℤ≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) |
157 | 146, 155,
156 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) |
158 | | fzsn 12700 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑀 + 1) ∈ ℤ →
((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)}) |
159 | 143, 158 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)}) |
160 | 159 | uneq1d 3989 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁))) |
161 | 157, 160 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁))) |
162 | 161 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) |
163 | 162 | uneq2d 3990 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}))) |
164 | | xpundir 5418 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) |
165 | | ovex 6954 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑀 + 1) ∈ V |
166 | | c0ex 10370 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 0 ∈
V |
167 | 165, 166 | xpsn 6672 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({(𝑀 + 1)} × {0}) =
{〈(𝑀 + 1),
0〉} |
168 | 167 | uneq1i 3986 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (({(𝑀 + 1)} × {0}) ∪
((((𝑀 + 1) + 1)...𝑁) × {0})) = ({〈(𝑀 + 1), 0〉} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) |
169 | 164, 168 | eqtri 2802 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({〈(𝑀 + 1), 0〉} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) |
170 | 169 | uneq2i 3987 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
({〈(𝑀 + 1), 0〉}
∪ ((((𝑀 + 1) +
1)...𝑁) ×
{0}))) |
171 | | unass 3993 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
{〈(𝑀 + 1), 0〉})
∪ ((((𝑀 + 1) +
1)...𝑁) × {0})) =
((𝑛 ∈ (1...𝑀) ↦ (((1st
‘𝑘)‘𝑛) + (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
({〈(𝑀 + 1), 0〉}
∪ ((((𝑀 + 1) +
1)...𝑁) ×
{0}))) |
172 | 170, 171 | eqtr4i 2805 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
{〈(𝑀 + 1), 0〉})
∪ ((((𝑀 + 1) +
1)...𝑁) ×
{0})) |
173 | 163, 172 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
{〈(𝑀 + 1), 0〉})
∪ ((((𝑀 + 1) +
1)...𝑁) ×
{0}))) |
174 | 173 | ad3antrrr 720 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
{〈(𝑀 + 1), 0〉})
∪ ((((𝑀 + 1) +
1)...𝑁) ×
{0}))) |
175 | 165 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ V) |
176 | 166 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ V) |
177 | 110 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1))) |
178 | 177 | ad3antrrr 720 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1))) |
179 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = (𝑀 + 1) → ((1st ‘𝑘)‘𝑛) = ((1st ‘𝑘)‘(𝑀 + 1))) |
180 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = (𝑀 + 1) → (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛) =
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))) |
181 | 179, 180 | oveq12d 6940 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = (𝑀 + 1) → (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛)) =
(((1st ‘𝑘)‘(𝑀 + 1)) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘(𝑀 +
1)))) |
182 | | simplrl 767 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((1st ‘𝑘)‘(𝑀 + 1)) = 0) |
183 | | imain 6219 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (Fun
◡(2nd ‘𝑘) → ((2nd
‘𝑘) “
((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))) |
184 | 76, 83, 183 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd
‘𝑘) “
((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))) |
185 | 184 | ad2antlr 717 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))) |
186 | | elfznn0 12751 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0) |
187 | 186 | nn0red 11703 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ) |
188 | 187 | ltp1d 11308 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1)) |
189 | | fzdisj 12685 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅) |
190 | 188, 189 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅) |
191 | 190 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd ‘𝑘) “
∅)) |
192 | | ima0 5735 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((2nd ‘𝑘) “ ∅) = ∅ |
193 | 191, 192 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) |
194 | 185, 193 | sylan9req 2835 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) |
195 | | simplr 759 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) |
196 | 92 | ad2antlr 717 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd ‘𝑘) Fn (1...(𝑀 + 1))) |
197 | | nn0p1nn 11683 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ ℕ0
→ (𝑗 + 1) ∈
ℕ) |
198 | | nnuz 12029 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ℕ =
(ℤ≥‘1) |
199 | 197, 198 | syl6eleq 2869 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ ℕ0
→ (𝑗 + 1) ∈
(ℤ≥‘1)) |
200 | | fzss1 12697 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑗 + 1) ∈
(ℤ≥‘1) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1))) |
201 | 186, 199,
200 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1))) |
202 | | elfzuz3 12656 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ≥‘𝑗)) |
203 | | eluzp1p1 12018 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑀 ∈
(ℤ≥‘𝑗) → (𝑀 + 1) ∈
(ℤ≥‘(𝑗 + 1))) |
204 | | eluzfz2 12666 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑀 + 1) ∈
(ℤ≥‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))) |
205 | 202, 203,
204 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))) |
206 | 201, 205 | jca 507 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))) |
207 | | fnfvima 6769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((2nd ‘𝑘) Fn (1...(𝑀 + 1)) ∧ ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))) → ((2nd ‘𝑘)‘(𝑀 + 1)) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
208 | 207 | 3expb 1110 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((2nd ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))) → ((2nd
‘𝑘)‘(𝑀 + 1)) ∈ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
209 | 196, 206,
208 | syl2an 589 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd ‘𝑘)‘(𝑀 + 1)) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
210 | 195, 209 | eqeltrrd 2860 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
211 | | 1ex 10372 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 1 ∈
V |
212 | | fnconstg 6343 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (1 ∈
V → (((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗))) |
213 | 211, 212 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗)) |
214 | | fnconstg 6343 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (0 ∈
V → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
215 | 166, 214 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) |
216 | | fvun2 6530 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗)) ∧
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn
((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ∧ ((((2nd
‘𝑘) “
(1...𝑗)) ∩
((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))) → (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘(𝑀 + 1)) =
((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1))) |
217 | 213, 215,
216 | mp3an12 1524 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) → (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘(𝑀 + 1)) =
((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1))) |
218 | 194, 210,
217 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘(𝑀 + 1)) =
((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1))) |
219 | 166 | fvconst2 6741 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑀 + 1) ∈ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0) |
220 | 210, 219 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0) |
221 | 218, 220 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘(𝑀 + 1)) =
0) |
222 | 221 | adantlrl 710 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘(𝑀 + 1)) =
0) |
223 | 182, 222 | oveq12d 6940 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘)‘(𝑀 + 1)) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘(𝑀 + 1))) = (0
+ 0)) |
224 | | 00id 10551 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0 + 0) =
0 |
225 | 223, 224 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘)‘(𝑀 + 1)) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘(𝑀 + 1))) =
0) |
226 | 181, 225 | sylan9eqr 2836 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛)) =
0) |
227 | 175, 176,
178, 226 | fmptapd 6704 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
{〈(𝑀 + 1), 0〉}) =
(𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st
‘𝑘)‘𝑛) + (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛)))) |
228 | 227 | uneq1d 3989 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
{〈(𝑀 + 1), 0〉})
∪ ((((𝑀 + 1) +
1)...𝑁) × {0})) =
((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st
‘𝑘)‘𝑛) + (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
((((𝑀 + 1) + 1)...𝑁) ×
{0}))) |
229 | 174, 228 | eqtrd 2814 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st
‘𝑘)‘𝑛) + (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
((((𝑀 + 1) + 1)...𝑁) ×
{0}))) |
230 | | elmapfn 8163 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) → (1st
‘𝑘) Fn (1...(𝑀 + 1))) |
231 | 61, 230 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st ‘𝑘) Fn (1...(𝑀 + 1))) |
232 | | fnssres 6250 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((1st ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st ‘𝑘) ↾ (1...𝑀)) Fn (1...𝑀)) |
233 | 231, 64, 232 | sylancl 580 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st
‘𝑘) ↾
(1...𝑀)) Fn (1...𝑀)) |
234 | 233 | ad3antlr 721 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1st ‘𝑘) ↾ (1...𝑀)) Fn (1...𝑀)) |
235 | | simplr 759 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) |
236 | | fnconstg 6343 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (0 ∈
V → (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀))) |
237 | 166, 236 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) |
238 | 213, 237 | pm3.2i 464 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗)) ∧
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}) Fn
((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀))) |
239 | | imain 6219 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (Fun
◡(2nd ‘𝑘) → ((2nd
‘𝑘) “
((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)))) |
240 | 76, 83, 239 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd
‘𝑘) “
((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)))) |
241 | | fzdisj 12685 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅) |
242 | 188, 241 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅) |
243 | 242 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((2nd ‘𝑘) “
∅)) |
244 | 243, 192 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅) |
245 | 240, 244 | sylan9req 2835 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀))) = ∅) |
246 | | fnun 6243 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗)) ∧
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}) Fn
((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀))) ∧ (((2nd
‘𝑘) “
(1...𝑗)) ∩
((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀))) = ∅) →
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd
‘𝑘) “
(1...𝑗)) ∪
((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)))) |
247 | 238, 245,
246 | sylancr 581 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) Fn
(((2nd ‘𝑘)
“ (1...𝑗)) ∪
((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)))) |
248 | 235, 247 | sylan 575 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) Fn
(((2nd ‘𝑘)
“ (1...𝑗)) ∪
((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)))) |
249 | 101 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd
‘𝑘) “ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) |
250 | 85 | ad3antlr 721 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd
‘𝑘) “
(1...(𝑀 + 1))) ∖
((2nd ‘𝑘)
“ {(𝑀 +
1)}))) |
251 | 186, 197 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℕ) |
252 | 251, 198 | syl6eleq 2869 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈
(ℤ≥‘1)) |
253 | | fzsplit2 12683 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝑗 + 1) ∈
(ℤ≥‘1) ∧ 𝑀 ∈ (ℤ≥‘𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) |
254 | 252, 202,
253 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) |
255 | 128, 254 | sylan9eq 2834 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) |
256 | 255 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd
‘𝑘) “
((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))) |
257 | 250, 256 | eqtr3d 2816 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd
‘𝑘) “ {(𝑀 + 1)})) = ((2nd
‘𝑘) “
((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))) |
258 | 125 | ad3antrrr 720 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀)) |
259 | 249, 257,
258 | 3eqtr3rd 2823 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) = ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))) |
260 | | imaundi 5799 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀))) |
261 | 259, 260 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)))) |
262 | 261 | fneq2d 6227 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ↔ ((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) Fn
(((2nd ‘𝑘)
“ (1...𝑗)) ∪
((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀))))) |
263 | 248, 262 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀)) |
264 | | fzss2 12698 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑀 ∈
(ℤ≥‘𝑗) → (1...𝑗) ⊆ (1...𝑀)) |
265 | | resima2 5681 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((1...𝑗) ⊆
(1...𝑀) →
(((2nd ‘𝑘)
↾ (1...𝑀)) “
(1...𝑗)) = ((2nd
‘𝑘) “
(1...𝑗))) |
266 | 202, 264,
265 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd ‘𝑘) “ (1...𝑗))) |
267 | 266 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) = (((2nd
‘𝑘) “
(1...𝑗)) ×
{1})) |
268 | 186, 199 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈
(ℤ≥‘1)) |
269 | | fzss1 12697 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑗 + 1) ∈
(ℤ≥‘1) → ((𝑗 + 1)...𝑀) ⊆ (1...𝑀)) |
270 | | resima2 5681 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑗 + 1)...𝑀) ⊆ (1...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))) |
271 | 268, 269,
270 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))) |
272 | 271 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) |
273 | 267, 272 | uneq12d 3991 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (0...𝑀) → (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) ×
{0}))) |
274 | 273 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) ×
{0}))) |
275 | 274 | fneq1d 6226 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ↔ ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))) |
276 | 263, 275 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀)) |
277 | | fzfid 13091 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) ∈ Fin) |
278 | | inidm 4043 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1...𝑀) ∩
(1...𝑀)) = (1...𝑀) |
279 | | fvres 6465 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ (1...𝑀) → (((1st ‘𝑘) ↾ (1...𝑀))‘𝑛) = ((1st ‘𝑘)‘𝑛)) |
280 | 279 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((1st ‘𝑘) ↾ (1...𝑀))‘𝑛) = ((1st ‘𝑘)‘𝑛)) |
281 | | disjsn 4478 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((1...𝑀) ∩
{(𝑀 + 1)}) = ∅ ↔
¬ (𝑀 + 1) ∈
(1...𝑀)) |
282 | 122, 281 | sylibr 226 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) |
283 | 282 | ad3antrrr 720 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) |
284 | 263, 283 | jca 507 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)) |
285 | | fnconstg 6343 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (0 ∈
V → ({(𝑀 + 1)} ×
{0}) Fn {(𝑀 +
1)}) |
286 | 166, 285 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)} |
287 | | fvun1 6529 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)) |
288 | 286, 287 | mp3an2 1522 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)) |
289 | 288 | anassrs 461 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)) |
290 | 284, 289 | sylan 575 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)) |
291 | 251 | nnzd 11833 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℤ) |
292 | 186 | nn0cnd 11704 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ) |
293 | | pncan1 10799 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗) |
294 | 292, 293 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1) − 1) = 𝑗) |
295 | 294 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑗 ∈ (0...𝑀) →
(ℤ≥‘((𝑗 + 1) − 1)) =
(ℤ≥‘𝑗)) |
296 | 202, 295 | eleqtrrd 2862 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈
(ℤ≥‘((𝑗 + 1) − 1))) |
297 | | fzsuc2 12716 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝑗 + 1) ∈ ℤ ∧ 𝑀 ∈
(ℤ≥‘((𝑗 + 1) − 1))) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) |
298 | 291, 296,
297 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) |
299 | 298 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd ‘𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))) |
300 | | imaundi 5799 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((2nd ‘𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) |
301 | 299, 300 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)}))) |
302 | 301 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) × {0})) |
303 | | xpundir 5418 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) × {0}) = ((((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) ×
{0})) |
304 | 302, 303 | syl6eq 2830 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) ×
{0}))) |
305 | 304 | uneq2d 3990 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) =
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) ×
{0})))) |
306 | | unass 3993 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) × {0})) =
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) ×
{0}))) |
307 | 305, 306 | syl6eqr 2832 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) =
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) ×
{0}))) |
308 | 307 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) =
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) ×
{0}))) |
309 | 98 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ({((2nd
‘𝑘)‘(𝑀 + 1))} × {0}) =
(((2nd ‘𝑘)
“ {(𝑀 + 1)}) ×
{0})) |
310 | 309 | uneq2d 3990 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪
({((2nd ‘𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪
(((2nd ‘𝑘)
“ {(𝑀 + 1)}) ×
{0}))) |
311 | 310 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪
({((2nd ‘𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪
(((2nd ‘𝑘)
“ {(𝑀 + 1)}) ×
{0}))) |
312 | 308, 311 | eqtr4d 2817 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) =
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd
‘𝑘)‘(𝑀 + 1))} ×
{0}))) |
313 | 99 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0}) = ({(𝑀 + 1)} × {0})) |
314 | 313 | uneq2d 3990 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪
({((2nd ‘𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} ×
{0}))) |
315 | 312, 314 | sylan9eq 2834 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) =
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))) |
316 | 315 | an32s 642 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) =
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))) |
317 | 316 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛) =
((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛)) |
318 | 317 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛) =
((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛)) |
319 | 273 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 ∈ (0...𝑀) → ((((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)) |
320 | 319 | ad2antlr 717 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)) |
321 | 290, 318,
320 | 3eqtr4rd 2825 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛)) |
322 | 234, 276,
277, 277, 278, 280, 321 | offval 7181 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 +
(((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) = (𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛)))) |
323 | 322 | uneq1d 3989 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 +
(((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(((𝑀 + 1)...𝑁) ×
{0}))) |
324 | 323 | adantlrl 710 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 +
(((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
(((𝑀 + 1)...𝑁) ×
{0}))) |
325 | | simplr 759 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) |
326 | 231 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (1st ‘𝑘) Fn (1...(𝑀 + 1))) |
327 | 213, 215 | pm3.2i 464 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗)) ∧
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn
((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1)))) |
328 | 184, 193 | sylan9req 2835 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) |
329 | | fnun 6243 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗)) ∧
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn
((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1)))) ∧ (((2nd
‘𝑘) “
(1...𝑗)) ∩
((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) →
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd
‘𝑘) “
(1...𝑗)) ∪
((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))))) |
330 | 327, 328,
329 | sylancr 581 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn
(((2nd ‘𝑘)
“ (1...𝑗)) ∪
((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))))) |
331 | | peano2uz 12047 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑀 ∈
(ℤ≥‘𝑗) → (𝑀 + 1) ∈
(ℤ≥‘𝑗)) |
332 | 202, 331 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈
(ℤ≥‘𝑗)) |
333 | | fzsplit2 12683 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑗 + 1) ∈
(ℤ≥‘1) ∧ (𝑀 + 1) ∈
(ℤ≥‘𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) |
334 | 268, 332,
333 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) |
335 | 334 | imaeq2d 5720 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = ((2nd
‘𝑘) “
((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))) |
336 | | imaundi 5799 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
337 | 335, 336 | syl6req 2831 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd ‘𝑘) “ (1...(𝑀 + 1)))) |
338 | 337, 89 | sylan9eqr 2836 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = (1...(𝑀 + 1))) |
339 | 338 | fneq2d 6227 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn
(((2nd ‘𝑘)
“ (1...𝑗)) ∪
((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1)))) ↔
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)))) |
340 | 330, 339 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn
(1...(𝑀 +
1))) |
341 | | fzfid 13091 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) ∈ Fin) |
342 | | inidm 4043 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1...(𝑀 + 1)) ∩
(1...(𝑀 + 1))) =
(1...(𝑀 +
1)) |
343 | | eqidd 2779 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → ((1st ‘𝑘)‘𝑛) = ((1st ‘𝑘)‘𝑛)) |
344 | | eqidd 2779 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛) =
(((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) |
345 | 326, 340,
341, 341, 342, 343, 344 | offval 7181 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((1st ‘𝑘) ∘𝑓 +
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st
‘𝑘)‘𝑛) + (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛)))) |
346 | 345 | uneq1d 3989 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘) ∘𝑓 +
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st
‘𝑘)‘𝑛) + (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
((((𝑀 + 1) + 1)...𝑁) ×
{0}))) |
347 | 325, 346 | sylan 575 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘) ∘𝑓 +
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st
‘𝑘)‘𝑛) + (((((2nd
‘𝑘) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑘)
“ ((𝑗 + 1)...(𝑀 + 1))) ×
{0}))‘𝑛))) ∪
((((𝑀 + 1) + 1)...𝑁) ×
{0}))) |
348 | 229, 324,
347 | 3eqtr4rd 2825 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘) ∘𝑓 +
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0}))) |
349 | 348 | csbeq1d 3758 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
350 | 349 | eqeq2d 2788 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
351 | 350 | rexbidva 3234 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
352 | 351 | ralbidv 3168 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
353 | 352 | biimpd 221 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
354 | 353 | impr 448 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
355 | 142, 354 | sylan2b 587 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
356 | | 1st2nd2 7484 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → 𝑘 = 〈(1st ‘𝑘), (2nd ‘𝑘)〉) |
357 | 356 | ad2antlr 717 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = 〈(1st ‘𝑘), (2nd ‘𝑘)〉) |
358 | | fnsnsplit 6721 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → (1st ‘𝑘) = (((1st
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((1st ‘𝑘)‘(𝑀 + 1))〉})) |
359 | 231, 96, 358 | syl2anr 590 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (1st
‘𝑘) =
(((1st ‘𝑘)
↾ ((1...(𝑀 + 1))
∖ {(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((1st ‘𝑘)‘(𝑀 + 1))〉})) |
360 | 359 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st
‘𝑘)‘(𝑀 + 1)) = 0) →
(1st ‘𝑘) =
(((1st ‘𝑘)
↾ ((1...(𝑀 + 1))
∖ {(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((1st ‘𝑘)‘(𝑀 + 1))〉})) |
361 | 125 | reseq2d 5642 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1st
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
((1st ‘𝑘)
↾ (1...𝑀))) |
362 | 361 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((1st
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
((1st ‘𝑘)
↾ (1...𝑀))) |
363 | | opeq2 4637 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑘)‘(𝑀 + 1)) = 0 → 〈(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))〉 = 〈(𝑀 + 1), 0〉) |
364 | 363 | sneqd 4410 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑘)‘(𝑀 + 1)) = 0 → {〈(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))〉} = {〈(𝑀 + 1), 0〉}) |
365 | | uneq12 3985 |
. . . . . . . . . . . . . . 15
⊢
((((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st ‘𝑘) ↾ (1...𝑀)) ∧ {〈(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))〉} = {〈(𝑀 + 1), 0〉}) → (((1st
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((1st ‘𝑘)‘(𝑀 + 1))〉}) = (((1st
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1),
0〉})) |
366 | 362, 364,
365 | syl2an 589 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st
‘𝑘)‘(𝑀 + 1)) = 0) →
(((1st ‘𝑘)
↾ ((1...(𝑀 + 1))
∖ {(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((1st ‘𝑘)‘(𝑀 + 1))〉}) = (((1st
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1),
0〉})) |
367 | 360, 366 | eqtrd 2814 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st
‘𝑘)‘(𝑀 + 1)) = 0) →
(1st ‘𝑘) =
(((1st ‘𝑘)
↾ (1...𝑀)) ∪
{〈(𝑀 + 1),
0〉})) |
368 | 367 | adantrr 707 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (1st ‘𝑘) = (((1st
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1),
0〉})) |
369 | | fnsnsplit 6721 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → (2nd ‘𝑘) = (((2nd
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((2nd ‘𝑘)‘(𝑀 + 1))〉})) |
370 | 92, 96, 369 | syl2anr 590 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (2nd
‘𝑘) =
(((2nd ‘𝑘)
↾ ((1...(𝑀 + 1))
∖ {(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((2nd ‘𝑘)‘(𝑀 + 1))〉})) |
371 | 370 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd ‘𝑘) = (((2nd
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((2nd ‘𝑘)‘(𝑀 + 1))〉})) |
372 | 125 | reseq2d 5642 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
((2nd ‘𝑘)
↾ (1...𝑀))) |
373 | 372 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((2nd
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
((2nd ‘𝑘)
↾ (1...𝑀))) |
374 | | opeq2 4637 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → 〈(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))〉 = 〈(𝑀 + 1), (𝑀 + 1)〉) |
375 | 374 | sneqd 4410 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {〈(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))〉} = {〈(𝑀 + 1), (𝑀 + 1)〉}) |
376 | | uneq12 3985 |
. . . . . . . . . . . . . . 15
⊢
((((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) ↾ (1...𝑀)) ∧ {〈(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))〉} = {〈(𝑀 + 1), (𝑀 + 1)〉}) → (((2nd
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) ∪
{〈(𝑀 + 1),
((2nd ‘𝑘)‘(𝑀 + 1))〉}) = (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
377 | 373, 375,
376 | syl2an 589 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {〈(𝑀 + 1), ((2nd
‘𝑘)‘(𝑀 + 1))〉}) =
(((2nd ‘𝑘)
↾ (1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
378 | 371, 377 | eqtrd 2814 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd ‘𝑘) = (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
379 | 378 | adantrl 706 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (2nd ‘𝑘) = (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
380 | 368, 379 | opeq12d 4644 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 〈(1st
‘𝑘), (2nd
‘𝑘)〉 =
〈(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉}), (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉) |
381 | 357, 380 | eqtrd 2814 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st
‘𝑘)‘(𝑀 + 1)) = 0 ∧
((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = 〈(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉}), (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉) |
382 | 381 | 3adantr1 1171 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = 〈(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉}), (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉) |
383 | | fvex 6459 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1st ‘𝑘) ∈ V |
384 | 383 | resex 5693 |
. . . . . . . . . . . . . . . . . 18
⊢
((1st ‘𝑘) ↾ (1...𝑀)) ∈ V |
385 | 384, 133 | op1std 7455 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(1st ‘𝑡) =
((1st ‘𝑘)
↾ (1...𝑀))) |
386 | 384, 133 | op2ndd 7456 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(2nd ‘𝑡) =
((2nd ‘𝑘)
↾ (1...𝑀))) |
387 | 386 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((2nd ‘𝑡)
“ (1...𝑗)) =
(((2nd ‘𝑘)
↾ (1...𝑀)) “
(1...𝑗))) |
388 | 387 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(((2nd ‘𝑡)
“ (1...𝑗)) ×
{1}) = ((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1})) |
389 | 386 | imaeq1d 5719 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((2nd ‘𝑡)
“ ((𝑗 + 1)...𝑀)) = (((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀))) |
390 | 389 | xpeq1d 5384 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(((2nd ‘𝑡)
“ ((𝑗 + 1)...𝑀)) × {0}) =
((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) |
391 | 388, 390 | uneq12d 3991 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0})) = (((((2nd
‘𝑘) ↾
(1...𝑀)) “ (1...𝑗)) × {1}) ∪
((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) |
392 | 385, 391 | oveq12d 6940 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((1st ‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) = (((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})))) |
393 | 392 | uneq1d 3989 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(((1st ‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0}))) |
394 | 393 | csbeq1d 3758 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
⦋(((1st ‘𝑡) ∘𝑓 +
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
395 | 394 | eqeq2d 2788 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(𝑖 =
⦋(((1st ‘𝑡) ∘𝑓 +
((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
396 | 395 | rexbidv 3237 |
. . . . . . . . . . . 12
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
397 | 396 | ralbidv 3168 |
. . . . . . . . . . 11
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(∀𝑖 ∈
(0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
398 | 385 | uneq1d 3989 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((1st ‘𝑡)
∪ {〈(𝑀 + 1),
0〉}) = (((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉})) |
399 | 386 | uneq1d 3989 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉}) =
(((2nd ‘𝑘)
↾ (1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
400 | 398, 399 | opeq12d 4644 |
. . . . . . . . . . . 12
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})〉 =
〈(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉}), (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉) |
401 | 400 | eqeq2d 2788 |
. . . . . . . . . . 11
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(𝑘 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
↔ 𝑘 =
〈(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉}), (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉)) |
402 | 397, 401 | anbi12d 624 |
. . . . . . . . . 10
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((∀𝑖 ∈
(0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
↔ (∀𝑖 ∈
(0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉}), (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉))) |
403 | 402 | rspcev 3511 |
. . . . . . . . 9
⊢
((〈((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))〉 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st
‘𝑘) ↾
(1...𝑀))
∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉}), (((2nd
‘𝑘) ↾
(1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})〉)) →
∃𝑡 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉)) |
404 | 139, 355,
382, 403 | syl12anc 827 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉)) |
405 | 404 | ex 403 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉))) |
406 | | elrabi 3567 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} → 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) |
407 | | elrabi 3567 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} → 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) |
408 | 406, 407 | anim12i 606 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∧ 𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) |
409 | | eqtr2 2800 |
. . . . . . . . . . . 12
⊢ ((𝑘 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})〉 =
〈((1st ‘𝑛) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉) |
410 | 22, 24 | opth 5176 |
. . . . . . . . . . . . 13
⊢
(〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})〉 =
〈((1st ‘𝑛) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})〉 ↔
(((1st ‘𝑡)
∪ {〈(𝑀 + 1),
0〉}) = ((1st ‘𝑛) ∪ {〈(𝑀 + 1), 0〉}) ∧ ((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉}))) |
411 | | difeq1 3944 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) = ((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉})
→ (((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) ∖ {〈(𝑀 + 1), 0〉}) =
(((1st ‘𝑛)
∪ {〈(𝑀 + 1),
0〉}) ∖ {〈(𝑀
+ 1), 0〉})) |
412 | | difun2 4272 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) ∖ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑡)
∖ {〈(𝑀 + 1),
0〉}) |
413 | | difun2 4272 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑛) ∪ {〈(𝑀 + 1), 0〉}) ∖ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑛)
∖ {〈(𝑀 + 1),
0〉}) |
414 | 411, 412,
413 | 3eqtr3g 2837 |
. . . . . . . . . . . . . 14
⊢
(((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) = ((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉})
→ ((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) = ((1st
‘𝑛) ∖
{〈(𝑀 + 1),
0〉})) |
415 | | difeq1 3944 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉}) →
(((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉}) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) =
(((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉}) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
416 | | difun2 4272 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑡) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) |
417 | | difun2 4272 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑛) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) |
418 | 415, 416,
417 | 3eqtr3g 2837 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉}) →
((2nd ‘𝑡)
∖ {〈(𝑀 + 1),
(𝑀 + 1)〉}) =
((2nd ‘𝑛)
∖ {〈(𝑀 + 1),
(𝑀 +
1)〉})) |
419 | 414, 418 | anim12i 606 |
. . . . . . . . . . . . 13
⊢
((((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) = ((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉})
∧ ((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) →
(((1st ‘𝑡)
∖ {〈(𝑀 + 1),
0〉}) = ((1st ‘𝑛) ∖ {〈(𝑀 + 1), 0〉}) ∧ ((2nd
‘𝑡) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}))) |
420 | 410, 419 | sylbi 209 |
. . . . . . . . . . . 12
⊢
(〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑡) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})〉 =
〈((1st ‘𝑛) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})〉 →
(((1st ‘𝑡)
∖ {〈(𝑀 + 1),
0〉}) = ((1st ‘𝑛) ∖ {〈(𝑀 + 1), 0〉}) ∧ ((2nd
‘𝑡) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}))) |
421 | 409, 420 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑘 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ (((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) = ((1st
‘𝑛) ∖
{〈(𝑀 + 1), 0〉})
∧ ((2nd ‘𝑡) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}))) |
422 | | elmapfn 8163 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st
‘𝑡) Fn (1...𝑀)) |
423 | | fnop 6240 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1st ‘𝑡) Fn (1...𝑀) ∧ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡)) → (𝑀 + 1) ∈ (1...𝑀)) |
424 | 423 | ex 403 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑡) Fn (1...𝑀) → (〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡) → (𝑀 + 1) ∈ (1...𝑀))) |
425 | 9, 422, 424 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡) → (𝑀 + 1) ∈ (1...𝑀))) |
426 | 425, 122 | nsyli 157 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡))) |
427 | 426 | impcom 398 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡)) |
428 | | difsn 4560 |
. . . . . . . . . . . . . . . 16
⊢ (¬
〈(𝑀 + 1), 0〉
∈ (1st ‘𝑡) → ((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) =
(1st ‘𝑡)) |
429 | 427, 428 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) =
(1st ‘𝑡)) |
430 | | xp1st 7477 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st ‘𝑛) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀))) |
431 | | elmapfn 8163 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑛) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st
‘𝑛) Fn (1...𝑀)) |
432 | | fnop 6240 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1st ‘𝑛) Fn (1...𝑀) ∧ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛)) → (𝑀 + 1) ∈ (1...𝑀)) |
433 | 432 | ex 403 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑛) Fn (1...𝑀) → (〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛) → (𝑀 + 1) ∈ (1...𝑀))) |
434 | 430, 431,
433 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛) → (𝑀 + 1) ∈ (1...𝑀))) |
435 | 434, 122 | nsyli 157 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛))) |
436 | 435 | impcom 398 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛)) |
437 | | difsn 4560 |
. . . . . . . . . . . . . . . 16
⊢ (¬
〈(𝑀 + 1), 0〉
∈ (1st ‘𝑛) → ((1st ‘𝑛) ∖ {〈(𝑀 + 1), 0〉}) =
(1st ‘𝑛)) |
438 | 436, 437 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((1st ‘𝑛) ∖ {〈(𝑀 + 1), 0〉}) =
(1st ‘𝑛)) |
439 | 429, 438 | eqeqan12d 2794 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) ∧ (𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑛)
∖ {〈(𝑀 + 1),
0〉}) ↔ (1st ‘𝑡) = (1st ‘𝑛))) |
440 | 439 | anandis 668 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑛)
∖ {〈(𝑀 + 1),
0〉}) ↔ (1st ‘𝑡) = (1st ‘𝑛))) |
441 | | f1ofn 6392 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd ‘𝑡) Fn (1...𝑀)) |
442 | | fnop 6240 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘𝑡) Fn (1...𝑀) ∧ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡)) → (𝑀 + 1) ∈ (1...𝑀)) |
443 | 442 | ex 403 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑡) Fn (1...𝑀) → (〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡) → (𝑀 + 1) ∈ (1...𝑀))) |
444 | 17, 441, 443 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡) → (𝑀 + 1) ∈ (1...𝑀))) |
445 | 444, 122 | nsyli 157 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡))) |
446 | 445 | impcom 398 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡)) |
447 | | difsn 4560 |
. . . . . . . . . . . . . . . 16
⊢ (¬
〈(𝑀 + 1), (𝑀 + 1)〉 ∈
(2nd ‘𝑡)
→ ((2nd ‘𝑡) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = (2nd ‘𝑡)) |
448 | 446, 447 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((2nd ‘𝑡) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = (2nd ‘𝑡)) |
449 | | xp2nd 7478 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd ‘𝑛) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) |
450 | | fvex 6459 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(2nd ‘𝑛) ∈ V |
451 | | f1oeq1 6380 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓 = (2nd ‘𝑛) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀))) |
452 | 450, 451 | elab 3558 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘𝑛) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀)) |
453 | 449, 452 | sylib 210 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀)) |
454 | | f1ofn 6392 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd ‘𝑛) Fn (1...𝑀)) |
455 | | fnop 6240 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘𝑛) Fn (1...𝑀) ∧ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛)) → (𝑀 + 1) ∈ (1...𝑀)) |
456 | 455 | ex 403 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑛) Fn (1...𝑀) → (〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛) → (𝑀 + 1) ∈ (1...𝑀))) |
457 | 453, 454,
456 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛) → (𝑀 + 1) ∈ (1...𝑀))) |
458 | 457, 122 | nsyli 157 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛))) |
459 | 458 | impcom 398 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛)) |
460 | | difsn 4560 |
. . . . . . . . . . . . . . . 16
⊢ (¬
〈(𝑀 + 1), (𝑀 + 1)〉 ∈
(2nd ‘𝑛)
→ ((2nd ‘𝑛) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = (2nd ‘𝑛)) |
461 | 459, 460 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((2nd ‘𝑛) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = (2nd ‘𝑛)) |
462 | 448, 461 | eqeqan12d 2794 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) ∧ (𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((2nd ‘𝑡) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) ↔
(2nd ‘𝑡) =
(2nd ‘𝑛))) |
463 | 462 | anandis 668 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((2nd ‘𝑡) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) ↔
(2nd ‘𝑡) =
(2nd ‘𝑛))) |
464 | 440, 463 | anbi12d 624 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑛)
∖ {〈(𝑀 + 1),
0〉}) ∧ ((2nd ‘𝑡) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉})) ↔
((1st ‘𝑡)
= (1st ‘𝑛)
∧ (2nd ‘𝑡) = (2nd ‘𝑛)))) |
465 | | xpopth 7486 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (((1st ‘𝑡) = (1st ‘𝑛) ∧ (2nd
‘𝑡) = (2nd
‘𝑛)) ↔ 𝑡 = 𝑛)) |
466 | 465 | adantl 475 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st ‘𝑡) = (1st ‘𝑛) ∧ (2nd
‘𝑡) = (2nd
‘𝑛)) ↔ 𝑡 = 𝑛)) |
467 | 464, 466 | bitrd 271 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑛)
∖ {〈(𝑀 + 1),
0〉}) ∧ ((2nd ‘𝑡) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉})) ↔ 𝑡 = 𝑛)) |
468 | 421, 467 | syl5ib 236 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛)) |
469 | 408, 468 | sylan2 586 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∧ 𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) → ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛)) |
470 | 469 | ralrimivva 3153 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛)) |
471 | 470 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛)) |
472 | 405, 471 | jctird 522 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
∧ ∀𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛)))) |
473 | | fveq2 6446 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑛 → (1st ‘𝑡) = (1st ‘𝑛)) |
474 | 473 | uneq1d 3989 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑛 → ((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑛)
∪ {〈(𝑀 + 1),
0〉})) |
475 | | fveq2 6446 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑛 → (2nd ‘𝑡) = (2nd ‘𝑛)) |
476 | 475 | uneq1d 3989 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑛 → ((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
477 | 474, 476 | opeq12d 4644 |
. . . . . . . . 9
⊢ (𝑡 = 𝑛 → 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 =
〈((1st ‘𝑛) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉) |
478 | 477 | eqeq2d 2788 |
. . . . . . . 8
⊢ (𝑡 = 𝑛 → (𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
↔ 𝑘 =
〈((1st ‘𝑛) ∪ {〈(𝑀 + 1), 0〉}), ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 +
1)〉})〉)) |
479 | 478 | reu4 3612 |
. . . . . . 7
⊢
(∃!𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
↔ (∃𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛))) |
480 | 58 | rexrab 3580 |
. . . . . . . 8
⊢
(∃𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
↔ ∃𝑡 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉)) |
481 | 480 | anbi1i 617 |
. . . . . . 7
⊢
((∃𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛)) ↔ (∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
∧ ∀𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛))) |
482 | 479, 481 | bitri 267 |
. . . . . 6
⊢
(∃!𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
↔ (∃𝑡 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑡)
∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
∧ ∀𝑡 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉 ∧
𝑘 = 〈((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑛)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉)
→ 𝑡 = 𝑛))) |
483 | 472, 482 | syl6ibr 244 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉)) |
484 | 483 | ralrimiva 3148 |
. . . 4
⊢ (𝜑 → ∀𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉)) |
485 | | fveq2 6446 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑘 → (1st ‘𝑠) = (1st ‘𝑘)) |
486 | | fveq2 6446 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑘 → (2nd ‘𝑠) = (2nd ‘𝑘)) |
487 | 486 | imaeq1d 5719 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd
‘𝑘) “
(1...𝑗))) |
488 | 487 | xpeq1d 5384 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) =
(((2nd ‘𝑘)
“ (1...𝑗)) ×
{1})) |
489 | 486 | imaeq1d 5719 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
490 | 489 | xpeq1d 5384 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) |
491 | 488, 490 | uneq12d 3991 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑘 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...(𝑀 + 1))) × {0})) =
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) |
492 | 485, 491 | oveq12d 6940 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑘 → ((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = ((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})))) |
493 | 492 | uneq1d 3989 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑘 → (((1st ‘𝑠) ∘𝑓 +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = (((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))) |
494 | 493 | csbeq1d 3758 |
. . . . . . . . 9
⊢ (𝑠 = 𝑘 → ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
495 | 494 | eqeq2d 2788 |
. . . . . . . 8
⊢ (𝑠 = 𝑘 → (𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
496 | 495 | rexbidv 3237 |
. . . . . . 7
⊢ (𝑠 = 𝑘 → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
497 | 496 | ralbidv 3168 |
. . . . . 6
⊢ (𝑠 = 𝑘 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
498 | 485 | fveq1d 6448 |
. . . . . . 7
⊢ (𝑠 = 𝑘 → ((1st ‘𝑠)‘(𝑀 + 1)) = ((1st ‘𝑘)‘(𝑀 + 1))) |
499 | 498 | eqeq1d 2780 |
. . . . . 6
⊢ (𝑠 = 𝑘 → (((1st ‘𝑠)‘(𝑀 + 1)) = 0 ↔ ((1st
‘𝑘)‘(𝑀 + 1)) = 0)) |
500 | 486 | fveq1d 6448 |
. . . . . . 7
⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠)‘(𝑀 + 1)) = ((2nd ‘𝑘)‘(𝑀 + 1))) |
501 | 500 | eqeq1d 2780 |
. . . . . 6
⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) |
502 | 497, 499,
501 | 3anbi123d 1509 |
. . . . 5
⊢ (𝑠 = 𝑘 → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)))) |
503 | 502 | ralrab 3578 |
. . . 4
⊢
(∀𝑘 ∈
{𝑠 ∈ (((0..^𝐾) ↑𝑚
(1...(𝑀 + 1))) ×
{𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
↔ ∀𝑘 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑘)
∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉)) |
504 | 484, 503 | sylibr 226 |
. . 3
⊢ (𝜑 → ∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉) |
505 | | eqid 2778 |
. . . 4
⊢ (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉) =
(𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉) |
506 | 505 | f1ompt 6645 |
. . 3
⊢ ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉):{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ (∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
∈ {𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ∧ ∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = 〈((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉)) |
507 | 60, 504, 506 | sylanbrc 578 |
. 2
⊢ (𝜑 → (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉):{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}) |
508 | | ovex 6954 |
. . . . 5
⊢
((0..^𝐾)
↑𝑚 (1...𝑀)) ∈ V |
509 | | ovex 6954 |
. . . . . 6
⊢
((1...𝑀)
↑𝑚 (1...𝑀)) ∈ V |
510 | | f1of 6391 |
. . . . . . . 8
⊢ (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑓:(1...𝑀)⟶(1...𝑀)) |
511 | 510 | ss2abi 3895 |
. . . . . . 7
⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ {𝑓 ∣ 𝑓:(1...𝑀)⟶(1...𝑀)} |
512 | 68, 68 | mapval 8152 |
. . . . . . 7
⊢
((1...𝑀)
↑𝑚 (1...𝑀)) = {𝑓 ∣ 𝑓:(1...𝑀)⟶(1...𝑀)} |
513 | 511, 512 | sseqtr4i 3857 |
. . . . . 6
⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ ((1...𝑀) ↑𝑚 (1...𝑀)) |
514 | 509, 513 | ssexi 5040 |
. . . . 5
⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ∈ V |
515 | 508, 514 | xpex 7240 |
. . . 4
⊢
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∈ V |
516 | 515 | rabex 5049 |
. . 3
⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ V |
517 | 516 | f1oen 8262 |
. 2
⊢ ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 +
1)〉})〉):{𝑠 ∈
(((0..^𝐾)
↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}) |
518 | 507, 517 | syl 17 |
1
⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st
‘𝑠)
∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd
‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}) |