Step | Hyp | Ref
| Expression |
1 | | poimir.0 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℕ) |
2 | 1 | adantr 484 |
. . . . . . 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 484 |
. . . . . . 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 484 |
. . . . . . 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 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 < 𝑁) |
9 | | xp1st 7746 |
. . . . . . . . 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 8459 |
. . . . . . . . 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 485 |
. . . . . . 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 7747 |
. . . . . . . . 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 6687 |
. . . . . . . . . 10
⊢
(2nd ‘𝑡) ∈ V |
15 | | f1oeq1 6606 |
. . . . . . . . . 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 3573 |
. . . . . . . . 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 221 |
. . . . . . . 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 485 |
. . . . . . 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 35403 |
. . . . . 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 6687 |
. . . . . . . . . . . . . . . 16
⊢
(1st ‘𝑡) ∈ V |
21 | | snex 5298 |
. . . . . . . . . . . . . . . 16
⊢
{〈(𝑀 + 1),
0〉} ∈ V |
22 | 20, 21 | unex 7487 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) ∈ V |
23 | | snex 5298 |
. . . . . . . . . . . . . . . 16
⊢
{〈(𝑀 + 1),
(𝑀 + 1)〉} ∈
V |
24 | 14, 23 | unex 7487 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) ∈ V |
25 | 22, 24 | op1std 7724 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (1st ‘𝑠) = ((1st ‘𝑡) ∪ {〈(𝑀 + 1),
0〉})) |
26 | 22, 24 | op2ndd 7725 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 = 〈((1st
‘𝑡) ∪
{〈(𝑀 + 1), 0〉}),
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉})〉
→ (2nd ‘𝑠) = ((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉})) |
27 | 26 | imaeq1d 5902 |
. . . . . . . . . . . . . . . 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 5554 |
. . . . . . . . . . . . . . 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 5902 |
. . . . . . . . . . . . . . . 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 5554 |
. . . . . . . . . . . . . . 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 4054 |
. . . . . . . . . . . . . 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 7188 |
. . . . . . . . . . . . 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 4052 |
. . . . . . . . . . . 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 3794 |
. . . . . . . . . . 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 2749 |
. . . . . . . . . 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 3207 |
. . . . . . . . 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 3109 |
. . . . . . . 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 6676 |
. . . . . . . . 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 2740 |
. . . . . . . 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 6676 |
. . . . . . . . 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 2740 |
. . . . . . . 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 1437 |
. . . . . . 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 3588 |
. . . . . 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 255 |
. . . . 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 3096 |
. . . 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 6674 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑡 → (1st ‘𝑠) = (1st ‘𝑡)) |
47 | | fveq2 6674 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑡 → (2nd ‘𝑠) = (2nd ‘𝑡)) |
48 | 47 | imaeq1d 5902 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd
‘𝑡) “
(1...𝑗))) |
49 | 48 | xpeq1d 5554 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) =
(((2nd ‘𝑡)
“ (1...𝑗)) ×
{1})) |
50 | 47 | imaeq1d 5902 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀))) |
51 | 50 | xpeq1d 5554 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd
‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0})) |
52 | 49, 51 | uneq12d 4054 |
. . . . . . . . . . 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 7188 |
. . . . . . . . . 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 4052 |
. . . . . . . . 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 3794 |
. . . . . . . 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 2749 |
. . . . . . 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 3207 |
. . . . . 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 3109 |
. . . . 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 3593 |
. . . 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 237 |
. . 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 7746 |
. . . . . . . . . . . . . . 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 8459 |
. . . . . . . . . . . . . . 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 13041 |
. . . . . . . . . . . . . 14
⊢
(1...𝑀) ⊆
(1...(𝑀 +
1)) |
65 | | fssres 6544 |
. . . . . . . . . . . . . 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 589 |
. . . . . . . . . . . . 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 7203 |
. . . . . . . . . . . . . 14
⊢
(0..^𝐾) ∈
V |
68 | | ovex 7203 |
. . . . . . . . . . . . . 14
⊢
(1...𝑀) ∈
V |
69 | 67, 68 | elmap 8481 |
. . . . . . . . . . . . 13
⊢
(((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑m (1...𝑀)) ↔ ((1st ‘𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾)) |
70 | 66, 69 | sylibr 237 |
. . . . . . . . . . . 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 727 |
. . . . . . . . . . 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 7747 |
. . . . . . . . . . . . . . . . 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 6687 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘𝑘) ∈ V |
74 | | f1oeq1 6606 |
. . . . . . . . . . . . . . . . . 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 3573 |
. . . . . . . . . . . . . . . . 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 221 |
. . . . . . . . . . . . . . . 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 6617 |
. . . . . . . . . . . . . . . 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 6632 |
. . . . . . . . . . . . . . 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 589 |
. . . . . . . . . . . . . 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 727 |
. . . . . . . . . . . . 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 6624 |
. . . . . . . . . . . . . . . . . . 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 500 |
. . . . . . . . . . . . . . . . . 18
⊢
((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → Fun ◡(2nd ‘𝑘)) |
84 | | imadif 6423 |
. . . . . . . . . . . . . . . . . 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 727 |
. . . . . . . . . . . . . . . 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 6625 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd ‘𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1))) |
88 | | foima 6597 |
. . . . . . . . . . . . . . . . . . 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 727 |
. . . . . . . . . . . . . . . . 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 6619 |
. . . . . . . . . . . . . . . . . . . 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 12015 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℕ0
→ (𝑀 + 1) ∈
ℕ) |
94 | 5, 93 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀 + 1) ∈ ℕ) |
95 | | elfz1end 13028 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 + 1) ∈ ℕ ↔
(𝑀 + 1) ∈ (1...(𝑀 + 1))) |
96 | 94, 95 | sylib 221 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1))) |
97 | | fnsnfv 6747 |
. . . . . . . . . . . . . . . . . . 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 600 |
. . . . . . . . . . . . . . . . . 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 4526 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {((2nd ‘𝑘)‘(𝑀 + 1))} = {(𝑀 + 1)}) |
100 | 98, 99 | sylan9req 2794 |
. . . . . . . . . . . . . . . . 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 4014 |
. . . . . . . . . . . . . . . 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 2773 |
. . . . . . . . . . . . . . 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 12093 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 1 ∈
ℤ |
104 | | nn0uz 12362 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
ℕ0 = (ℤ≥‘0) |
105 | | 1m1e0 11788 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (1
− 1) = 0 |
106 | 105 | fveq2i 6677 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(ℤ≥‘(1 − 1)) =
(ℤ≥‘0) |
107 | 104, 106 | eqtr4i 2764 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
ℕ0 = (ℤ≥‘(1 −
1)) |
108 | 5, 107 | eleqtrdi 2843 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(1
− 1))) |
109 | | fzsuc2 13056 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((1
∈ ℤ ∧ 𝑀
∈ (ℤ≥‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)})) |
110 | 103, 108,
109 | sylancr 590 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)})) |
111 | 110 | difeq1d 4012 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)})) |
112 | | difun2 4370 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...𝑀) ∪
{(𝑀 + 1)}) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)}) |
113 | 111, 112 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)})) |
114 | 5 | nn0red 12037 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑀 ∈ ℝ) |
115 | | ltp1 11558 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀 ∈ ℝ → 𝑀 < (𝑀 + 1)) |
116 | | peano2re 10891 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈
ℝ) |
117 | | ltnle 10798 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ) →
(𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀)) |
118 | 116, 117 | mpdan 687 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀 ∈ ℝ → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀)) |
119 | 115, 118 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℝ → ¬
(𝑀 + 1) ≤ 𝑀) |
120 | 114, 119 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀) |
121 | | elfzle2 13002 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀) |
122 | 120, 121 | nsyl 142 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀)) |
123 | | difsn 4686 |
. . . . . . . . . . . . . . . . . . 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 2773 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀)) |
126 | 125 | imaeq2d 5903 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑘) “
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
((2nd ‘𝑘)
“ (1...𝑀))) |
127 | 126 | ad2antrr 726 |
. . . . . . . . . . . . . . 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 726 |
. . . . . . . . . . . . . . 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 2781 |
. . . . . . . . . . . . . 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 | 129 | f1oeq3d 6615 |
. . . . . . . . . . . . 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))) |
131 | 81, 130 | mpbid 235 |
. . . . . . . . . . . 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)) |
132 | 73 | resex 5873 |
. . . . . . . . . . . . 13
⊢
((2nd ‘𝑘) ↾ (1...𝑀)) ∈ V |
133 | | f1oeq1 6606 |
. . . . . . . . . . . . 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))) |
134 | 132, 133 | elab 3573 |
. . . . . . . . . . . 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)) |
135 | 131, 134 | sylibr 237 |
. . . . . . . . . . 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)}) |
136 | 71, 135 | opelxpd 5563 |
. . . . . . . . . 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)})) |
137 | 136 | 3ad2antr3 1191 |
. . . . . . . . 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)})) |
138 | | 3anass 1096 |
. . . . . . . . . . 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)))) |
139 | 138 | biancomi 466 |
. . . . . . . . . 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)) |
140 | 94 | nnzd 12167 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
141 | | uzid 12339 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑀 + 1) ∈ ℤ →
(𝑀 + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
142 | | peano2uz 12383 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑀 + 1) ∈
(ℤ≥‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
143 | 140, 141,
142 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((𝑀 + 1) + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
144 | 5 | nn0zd 12166 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑀 ∈ ℤ) |
145 | 1 | nnzd 12167 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑁 ∈ ℤ) |
146 | | zltp1le 12113 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
147 | | peano2z 12104 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈
ℤ) |
148 | | eluz 12338 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁)) |
149 | 147, 148 | sylan 583 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁)) |
150 | 146, 149 | bitr4d 285 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) |
151 | 144, 145,
150 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) |
152 | 7, 151 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) |
153 | | fzsplit2 13023 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 + 1) + 1) ∈
(ℤ≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) |
154 | 143, 152,
153 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) |
155 | | fzsn 13040 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑀 + 1) ∈ ℤ →
((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)}) |
156 | 140, 155 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)}) |
157 | 156 | uneq1d 4052 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁))) |
158 | 154, 157 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁))) |
159 | 158 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) |
160 | 159 | uneq2d 4053 |
. . . . . . . . . . . . . . . . . . . 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}))) |
161 | | xpundir 5592 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) |
162 | | ovex 7203 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑀 + 1) ∈ V |
163 | | c0ex 10713 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 0 ∈
V |
164 | 162, 163 | xpsn 6913 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({(𝑀 + 1)} × {0}) =
{〈(𝑀 + 1),
0〉} |
165 | 164 | uneq1i 4049 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (({(𝑀 + 1)} × {0}) ∪
((((𝑀 + 1) + 1)...𝑁) × {0})) = ({〈(𝑀 + 1), 0〉} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) |
166 | 161, 165 | eqtri 2761 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({〈(𝑀 + 1), 0〉} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) |
167 | 166 | uneq2i 4050 |
. . . . . . . . . . . . . . . . . . . . 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}))) |
168 | | unass 4056 |
. . . . . . . . . . . . . . . . . . . . 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}))) |
169 | 167, 168 | eqtr4i 2764 |
. . . . . . . . . . . . . . . . . . . 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})) |
170 | 160, 169 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . . 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}))) |
171 | 170 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . 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}))) |
172 | 162 | 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) |
173 | 163 | 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) |
174 | 110 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1))) |
175 | 174 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . 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))) |
176 | | fveq2 6674 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = (𝑀 + 1) → ((1st ‘𝑘)‘𝑛) = ((1st ‘𝑘)‘(𝑀 + 1))) |
177 | | fveq2 6674 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
178 | 176, 177 | oveq12d 7188 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
179 | | simplrl 777 |
. . . . . . . . . . . . . . . . . . . . . . 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) |
180 | | imain 6424 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (Fun
◡(2nd ‘𝑘) → ((2nd
‘𝑘) “
((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))) |
181 | 76, 83, 180 | 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))))) |
182 | 181 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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))))) |
183 | | elfznn0 13091 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0) |
184 | 183 | nn0red 12037 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ) |
185 | 184 | ltp1d 11648 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1)) |
186 | | fzdisj 13025 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅) |
187 | 185, 186 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅) |
188 | 187 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd ‘𝑘) “
∅)) |
189 | | ima0 5919 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((2nd ‘𝑘) “ ∅) = ∅ |
190 | 188, 189 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) |
191 | 182, 190 | sylan9req 2794 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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)))) = ∅) |
192 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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)) |
193 | 92 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) |
194 | | nn0p1nn 12015 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ ℕ0
→ (𝑗 + 1) ∈
ℕ) |
195 | | nnuz 12363 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ℕ =
(ℤ≥‘1) |
196 | 194, 195 | eleqtrdi 2843 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ ℕ0
→ (𝑗 + 1) ∈
(ℤ≥‘1)) |
197 | | fzss1 13037 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑗 + 1) ∈
(ℤ≥‘1) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1))) |
198 | 183, 196,
197 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1))) |
199 | | elfzuz3 12995 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ≥‘𝑗)) |
200 | | eluzp1p1 12352 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑀 ∈
(ℤ≥‘𝑗) → (𝑀 + 1) ∈
(ℤ≥‘(𝑗 + 1))) |
201 | | eluzfz2 13006 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑀 + 1) ∈
(ℤ≥‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))) |
202 | 199, 200,
201 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))) |
203 | 198, 202 | jca 515 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))) |
204 | | fnfvima 7006 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) |
205 | 204 | 3expb 1121 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) |
206 | 193, 203,
205 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))) |
207 | 192, 206 | eqeltrrd 2834 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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)))) |
208 | | 1ex 10715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 1 ∈
V |
209 | | fnconstg 6566 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (1 ∈
V → (((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗))) |
210 | 208, 209 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗)) |
211 | | fnconstg 6566 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (0 ∈
V → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
212 | 163, 211 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) |
213 | | fvun2 6760 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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))) |
214 | 210, 212,
213 | mp3an12 1452 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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))) |
215 | 191, 207,
214 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . . . 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))) |
216 | 163 | fvconst2 6976 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑀 + 1) ∈ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0) |
217 | 207, 216 | 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) |
218 | 215, 217 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . . . . . 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) |
219 | 218 | adantlrl 720 |
. . . . . . . . . . . . . . . . . . . . . . 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) |
220 | 179, 219 | oveq12d 7188 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
221 | | 00id 10893 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0 + 0) =
0 |
222 | 220, 221 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . . . . 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) |
223 | 178, 222 | sylan9eqr 2795 |
. . . . . . . . . . . . . . . . . . . 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) |
224 | 172, 173,
175, 223 | fmptapd 6943 |
. . . . . . . . . . . . . . . . . . 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)))) |
225 | 224 | uneq1d 4052 |
. . . . . . . . . . . . . . . . . 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}))) |
226 | 171, 225 | eqtrd 2773 |
. . . . . . . . . . . . . . . . 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}))) |
227 | | elmapfn 8475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑘) ∈ ((0..^𝐾) ↑m (1...(𝑀 + 1))) → (1st ‘𝑘) Fn (1...(𝑀 + 1))) |
228 | 61, 227 | 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))) |
229 | | fnssres 6459 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((1st ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st ‘𝑘) ↾ (1...𝑀)) Fn (1...𝑀)) |
230 | 228, 64, 229 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . . 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)) |
231 | 230 | ad3antlr 731 |
. . . . . . . . . . . . . . . . . . . 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)) |
232 | | fnconstg 6566 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (0 ∈
V → (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀))) |
233 | 163, 232 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) |
234 | 210, 233 | pm3.2i 474 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd
‘𝑘) “
(1...𝑗)) ∧
(((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀)) × {0}) Fn
((2nd ‘𝑘)
“ ((𝑗 + 1)...𝑀))) |
235 | | imain 6424 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (Fun
◡(2nd ‘𝑘) → ((2nd
‘𝑘) “
((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)))) |
236 | 76, 83, 235 | 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)))) |
237 | | fzdisj 13025 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅) |
238 | 185, 237 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅) |
239 | 238 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((2nd ‘𝑘) “
∅)) |
240 | 239, 189 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅) |
241 | 236, 240 | sylan9req 2794 |
. . . . . . . . . . . . . . . . . . . . . . . 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))) = ∅) |
242 | | fnun 6449 |
. . . . . . . . . . . . . . . . . . . . . . . 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)))) |
243 | 234, 241,
242 | sylancr 590 |
. . . . . . . . . . . . . . . . . . . . . . 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)))) |
244 | 243 | ad4ant24 754 |
. . . . . . . . . . . . . . . . . . . . . 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)))) |
245 | 101 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . 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)})) |
246 | 85 | ad3antlr 731 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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)}))) |
247 | 183, 194 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℕ) |
248 | 247, 195 | eleqtrdi 2843 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈
(ℤ≥‘1)) |
249 | | fzsplit2 13023 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝑗 + 1) ∈
(ℤ≥‘1) ∧ 𝑀 ∈ (ℤ≥‘𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) |
250 | 248, 199,
249 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) |
251 | 128, 250 | sylan9eq 2793 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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))) |
252 | 251 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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)))) |
253 | 246, 252 | eqtr3d 2775 |
. . . . . . . . . . . . . . . . . . . . . . . . 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)))) |
254 | 125 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . 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)) |
255 | 245, 253,
254 | 3eqtr3rd 2782 |
. . . . . . . . . . . . . . . . . . . . . . . 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)))) |
256 | | imaundi 5982 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀))) |
257 | 255, 256 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . . . . . . 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)))) |
258 | 257 | fneq2d 6432 |
. . . . . . . . . . . . . . . . . . . . . 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))))) |
259 | 244, 258 | mpbird 260 |
. . . . . . . . . . . . . . . . . . . . 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)) |
260 | | fzss2 13038 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑀 ∈
(ℤ≥‘𝑗) → (1...𝑗) ⊆ (1...𝑀)) |
261 | | resima2 5860 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((1...𝑗) ⊆
(1...𝑀) →
(((2nd ‘𝑘)
↾ (1...𝑀)) “
(1...𝑗)) = ((2nd
‘𝑘) “
(1...𝑗))) |
262 | 199, 260,
261 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd ‘𝑘) “ (1...𝑗))) |
263 | 262 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) = (((2nd
‘𝑘) “
(1...𝑗)) ×
{1})) |
264 | 183, 196 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈
(ℤ≥‘1)) |
265 | | fzss1 13037 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑗 + 1) ∈
(ℤ≥‘1) → ((𝑗 + 1)...𝑀) ⊆ (1...𝑀)) |
266 | | resima2 5860 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑗 + 1)...𝑀) ⊆ (1...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))) |
267 | 264, 265,
266 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))) |
268 | 267 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) |
269 | 263, 268 | uneq12d 4054 |
. . . . . . . . . . . . . . . . . . . . . . 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}))) |
270 | 269 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . 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}))) |
271 | 270 | fneq1d 6431 |
. . . . . . . . . . . . . . . . . . . . 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))) |
272 | 259, 271 | mpbird 260 |
. . . . . . . . . . . . . . . . . . . 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)) |
273 | | fzfid 13432 |
. . . . . . . . . . . . . . . . . . . 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) |
274 | | inidm 4109 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1...𝑀) ∩
(1...𝑀)) = (1...𝑀) |
275 | | fvres 6693 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ (1...𝑀) → (((1st ‘𝑘) ↾ (1...𝑀))‘𝑛) = ((1st ‘𝑘)‘𝑛)) |
276 | 275 | adantl 485 |
. . . . . . . . . . . . . . . . . . . 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)) |
277 | | disjsn 4602 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((1...𝑀) ∩
{(𝑀 + 1)}) = ∅ ↔
¬ (𝑀 + 1) ∈
(1...𝑀)) |
278 | 122, 277 | sylibr 237 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) |
279 | 278 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . 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)}) = ∅) |
280 | 259, 279 | jca 515 |
. . . . . . . . . . . . . . . . . . . . . 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)}) = ∅)) |
281 | | fnconstg 6566 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (0 ∈
V → ({(𝑀 + 1)} ×
{0}) Fn {(𝑀 +
1)}) |
282 | 163, 281 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)} |
283 | | fvun1 6759 |
. . . . . . . . . . . . . . . . . . . . . . . 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)) |
284 | 282, 283 | mp3an2 1450 |
. . . . . . . . . . . . . . . . . . . . . . 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)) |
285 | 284 | anassrs 471 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
286 | 280, 285 | sylan 583 |
. . . . . . . . . . . . . . . . . . . . 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)) |
287 | 247 | nnzd 12167 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℤ) |
288 | 183 | nn0cnd 12038 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ) |
289 | | pncan1 11142 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗) |
290 | 288, 289 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1) − 1) = 𝑗) |
291 | 290 | fveq2d 6678 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑗 ∈ (0...𝑀) →
(ℤ≥‘((𝑗 + 1) − 1)) =
(ℤ≥‘𝑗)) |
292 | 199, 291 | eleqtrrd 2836 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈
(ℤ≥‘((𝑗 + 1) − 1))) |
293 | | fzsuc2 13056 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝑗 + 1) ∈ ℤ ∧ 𝑀 ∈
(ℤ≥‘((𝑗 + 1) − 1))) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) |
294 | 287, 292,
293 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) |
295 | 294 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd ‘𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))) |
296 | | imaundi 5982 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((2nd ‘𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) |
297 | 295, 296 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)}))) |
298 | 297 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) × {0})) |
299 | | xpundir 5592 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) × {0}) = ((((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) ×
{0})) |
300 | 298, 299 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd
‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd
‘𝑘) “ {(𝑀 + 1)}) ×
{0}))) |
301 | 300 | uneq2d 4053 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))) |
302 | | unass 4056 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))) |
303 | 301, 302 | eqtr4di 2791 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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}))) |
304 | 303 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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}))) |
305 | 98 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) |
306 | 305 | uneq2d 4053 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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}))) |
307 | 306 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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}))) |
308 | 304, 307 | eqtr4d 2776 |
. . . . . . . . . . . . . . . . . . . . . . . . 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}))) |
309 | 99 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0}) = ({(𝑀 + 1)} × {0})) |
310 | 309 | uneq2d 4053 |
. . . . . . . . . . . . . . . . . . . . . . . . 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}))) |
311 | 308, 310 | sylan9eq 2793 |
. . . . . . . . . . . . . . . . . . . . . . . 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}))) |
312 | 311 | an32s 652 |
. . . . . . . . . . . . . . . . . . . . . . 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}))) |
313 | 312 | fveq1d 6676 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
314 | 313 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 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)) |
315 | 269 | fveq1d 6676 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
316 | 315 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . 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)) |
317 | 286, 314,
316 | 3eqtr4rd 2784 |
. . . . . . . . . . . . . . . . . . . 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)) |
318 | 231, 272,
273, 273, 274, 276, 317 | offval 7433 |
. . . . . . . . . . . . . . . . . . 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)))) |
319 | 318 | uneq1d 4052 |
. . . . . . . . . . . . . . . . . 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}))) |
320 | 319 | adantlrl 720 |
. . . . . . . . . . . . . . . . 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}))) |
321 | 228 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 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))) |
322 | 210, 212 | pm3.2i 474 |
. . . . . . . . . . . . . . . . . . . . . 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)))) |
323 | 181, 190 | sylan9req 2794 |
. . . . . . . . . . . . . . . . . . . . . 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)))) = ∅) |
324 | | fnun 6449 |
. . . . . . . . . . . . . . . . . . . . . 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))))) |
325 | 322, 323,
324 | sylancr 590 |
. . . . . . . . . . . . . . . . . . . . 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))))) |
326 | | peano2uz 12383 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑀 ∈
(ℤ≥‘𝑗) → (𝑀 + 1) ∈
(ℤ≥‘𝑗)) |
327 | 199, 326 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈
(ℤ≥‘𝑗)) |
328 | | fzsplit2 13023 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑗 + 1) ∈
(ℤ≥‘1) ∧ (𝑀 + 1) ∈
(ℤ≥‘𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) |
329 | 264, 327,
328 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) |
330 | 329 | imaeq2d 5903 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = ((2nd
‘𝑘) “
((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))) |
331 | | imaundi 5982 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
332 | 330, 331 | eqtr2di 2790 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd ‘𝑘) “ (1...(𝑀 + 1)))) |
333 | 332, 89 | sylan9eqr 2795 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
334 | 333 | fneq2d 6432 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
335 | 325, 334 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . 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))) |
336 | | fzfid 13432 |
. . . . . . . . . . . . . . . . . . . 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) |
337 | | inidm 4109 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1...(𝑀 + 1)) ∩
(1...(𝑀 + 1))) =
(1...(𝑀 +
1)) |
338 | | eqidd 2739 |
. . . . . . . . . . . . . . . . . . . 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)) |
339 | | eqidd 2739 |
. . . . . . . . . . . . . . . . . . . 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)) |
340 | 321, 335,
336, 336, 337, 338, 339 | offval 7433 |
. . . . . . . . . . . . . . . . . . 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)))) |
341 | 340 | uneq1d 4052 |
. . . . . . . . . . . . . . . . . 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}))) |
342 | 341 | ad4ant24 754 |
. . . . . . . . . . . . . . . . 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}))) |
343 | 226, 320,
342 | 3eqtr4rd 2784 |
. . . . . . . . . . . . . . . 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}))) |
344 | 343 | csbeq1d 3794 |
. . . . . . . . . . . . . . 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) |
345 | 344 | eqeq2d 2749 |
. . . . . . . . . . . . . 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)) |
346 | 345 | rexbidva 3206 |
. . . . . . . . . . . . 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)) |
347 | 346 | ralbidv 3109 |
. . . . . . . . . . . 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)) |
348 | 347 | biimpd 232 |
. . . . . . . . . . 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)) |
349 | 348 | impr 458 |
. . . . . . . . . 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) |
350 | 139, 349 | sylan2b 597 |
. . . . . . . . 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) |
351 | | 1st2nd2 7753 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → 𝑘 = 〈(1st ‘𝑘), (2nd ‘𝑘)〉) |
352 | 351 | ad2antlr 727 |
. . . . . . . . . . 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)〉) |
353 | | fnsnsplit 6956 |
. . . . . . . . . . . . . . . 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))〉})) |
354 | 228, 96, 353 | syl2anr 600 |
. . . . . . . . . . . . . . 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))〉})) |
355 | 354 | adantr 484 |
. . . . . . . . . . . . . 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))〉})) |
356 | 125 | reseq2d 5825 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1st
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
((1st ‘𝑘)
↾ (1...𝑀))) |
357 | 356 | adantr 484 |
. . . . . . . . . . . . . . 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))) |
358 | | opeq2 4761 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑘)‘(𝑀 + 1)) = 0 → 〈(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))〉 = 〈(𝑀 + 1), 0〉) |
359 | 358 | sneqd 4528 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑘)‘(𝑀 + 1)) = 0 → {〈(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))〉} = {〈(𝑀 + 1), 0〉}) |
360 | | uneq12 4048 |
. . . . . . . . . . . . . . 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〉})) |
361 | 357, 359,
360 | syl2an 599 |
. . . . . . . . . . . . . 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〉})) |
362 | 355, 361 | eqtrd 2773 |
. . . . . . . . . . . . 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〉})) |
363 | 362 | adantrr 717 |
. . . . . . . . . . . 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〉})) |
364 | | fnsnsplit 6956 |
. . . . . . . . . . . . . . . 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))〉})) |
365 | 92, 96, 364 | syl2anr 600 |
. . . . . . . . . . . . . . 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))〉})) |
366 | 365 | adantr 484 |
. . . . . . . . . . . . . 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))〉})) |
367 | 125 | reseq2d 5825 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2nd
‘𝑘) ↾
((1...(𝑀 + 1)) ∖
{(𝑀 + 1)})) =
((2nd ‘𝑘)
↾ (1...𝑀))) |
368 | 367 | adantr 484 |
. . . . . . . . . . . . . . 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))) |
369 | | opeq2 4761 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → 〈(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))〉 = 〈(𝑀 + 1), (𝑀 + 1)〉) |
370 | 369 | sneqd 4528 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {〈(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))〉} = {〈(𝑀 + 1), (𝑀 + 1)〉}) |
371 | | uneq12 4048 |
. . . . . . . . . . . . . . 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)〉})) |
372 | 368, 370,
371 | syl2an 599 |
. . . . . . . . . . . . . 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)〉})) |
373 | 366, 372 | eqtrd 2773 |
. . . . . . . . . . . . 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)〉})) |
374 | 373 | adantrl 716 |
. . . . . . . . . . . 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)〉})) |
375 | 363, 374 | opeq12d 4769 |
. . . . . . . . . . 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)〉})〉) |
376 | 352, 375 | eqtrd 2773 |
. . . . . . . . . 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)〉})〉) |
377 | 376 | 3adantr1 1170 |
. . . . . . . . 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)〉})〉) |
378 | | fvex 6687 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1st ‘𝑘) ∈ V |
379 | 378 | resex 5873 |
. . . . . . . . . . . . . . . . . 18
⊢
((1st ‘𝑘) ↾ (1...𝑀)) ∈ V |
380 | 379, 132 | op1std 7724 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(1st ‘𝑡) =
((1st ‘𝑘)
↾ (1...𝑀))) |
381 | 379, 132 | op2ndd 7725 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(2nd ‘𝑡) =
((2nd ‘𝑘)
↾ (1...𝑀))) |
382 | 381 | imaeq1d 5902 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((2nd ‘𝑡)
“ (1...𝑗)) =
(((2nd ‘𝑘)
↾ (1...𝑀)) “
(1...𝑗))) |
383 | 382 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(((2nd ‘𝑡)
“ (1...𝑗)) ×
{1}) = ((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1})) |
384 | 381 | imaeq1d 5902 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((2nd ‘𝑡)
“ ((𝑗 + 1)...𝑀)) = (((2nd
‘𝑘) ↾
(1...𝑀)) “ ((𝑗 + 1)...𝑀))) |
385 | 384 | xpeq1d 5554 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
(((2nd ‘𝑡)
“ ((𝑗 + 1)...𝑀)) × {0}) =
((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) |
386 | 383, 385 | uneq12d 4054 |
. . . . . . . . . . . . . . . . 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}))) |
387 | 380, 386 | oveq12d 7188 |
. . . . . . . . . . . . . . . 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})))) |
388 | 387 | uneq1d 4052 |
. . . . . . . . . . . . . . 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}))) |
389 | 388 | csbeq1d 3794 |
. . . . . . . . . . . . . 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) |
390 | 389 | eqeq2d 2749 |
. . . . . . . . . . . . 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)) |
391 | 390 | rexbidv 3207 |
. . . . . . . . . . . 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)) |
392 | 391 | ralbidv 3109 |
. . . . . . . . . . 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)) |
393 | 380 | uneq1d 4052 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((1st ‘𝑡)
∪ {〈(𝑀 + 1),
0〉}) = (((1st ‘𝑘) ↾ (1...𝑀)) ∪ {〈(𝑀 + 1), 0〉})) |
394 | 381 | uneq1d 4052 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 〈((1st
‘𝑘) ↾
(1...𝑀)), ((2nd
‘𝑘) ↾
(1...𝑀))〉 →
((2nd ‘𝑡)
∪ {〈(𝑀 + 1),
(𝑀 + 1)〉}) =
(((2nd ‘𝑘)
↾ (1...𝑀)) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
395 | 393, 394 | opeq12d 4769 |
. . . . . . . . . . . 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)〉})〉) |
396 | 395 | eqeq2d 2749 |
. . . . . . . . . . 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)〉})〉)) |
397 | 392, 396 | anbi12d 634 |
. . . . . . . . . 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)〉})〉))) |
398 | 397 | rspcev 3526 |
. . . . . . . . 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)〉})〉)) |
399 | 137, 350,
377, 398 | syl12anc 836 |
. . . . . . . 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)〉})〉)) |
400 | 399 | ex 416 |
. . . . . . 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)〉})〉))) |
401 | | elrabi 3582 |
. . . . . . . . . . 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)})) |
402 | | elrabi 3582 |
. . . . . . . . . . 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)})) |
403 | 401, 402 | anim12i 616 |
. . . . . . . . . 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)}))) |
404 | | eqtr2 2759 |
. . . . . . . . . . . 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)〉})〉) |
405 | 22, 24 | opth 5334 |
. . . . . . . . . . . . 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)〉}))) |
406 | | difeq1 4006 |
. . . . . . . . . . . . . . 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〉})) |
407 | | difun2 4370 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) ∖ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑡)
∖ {〈(𝑀 + 1),
0〉}) |
408 | | difun2 4370 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑛) ∪ {〈(𝑀 + 1), 0〉}) ∖ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑛)
∖ {〈(𝑀 + 1),
0〉}) |
409 | 406, 407,
408 | 3eqtr3g 2796 |
. . . . . . . . . . . . . 14
⊢
(((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) = ((1st
‘𝑛) ∪
{〈(𝑀 + 1), 0〉})
→ ((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) = ((1st
‘𝑛) ∖
{〈(𝑀 + 1),
0〉})) |
410 | | difeq1 4006 |
. . . . . . . . . . . . . . 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)〉})) |
411 | | difun2 4370 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑡) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) |
412 | | difun2 4370 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑛) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∖
{〈(𝑀 + 1), (𝑀 + 1)〉}) |
413 | 410, 411,
412 | 3eqtr3g 2796 |
. . . . . . . . . . . . . 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)〉})) |
414 | 409, 413 | anim12i 616 |
. . . . . . . . . . . . 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)〉}))) |
415 | 405, 414 | sylbi 220 |
. . . . . . . . . . . 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)〉}))) |
416 | 404, 415 | 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)〉}))) |
417 | | elmapfn 8475 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑡) ∈ ((0..^𝐾) ↑m (1...𝑀)) → (1st ‘𝑡) Fn (1...𝑀)) |
418 | | fnop 6446 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1st ‘𝑡) Fn (1...𝑀) ∧ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡)) → (𝑀 + 1) ∈ (1...𝑀)) |
419 | 418 | ex 416 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑡) Fn (1...𝑀) → (〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡) → (𝑀 + 1) ∈ (1...𝑀))) |
420 | 9, 417, 419 | 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))) |
421 | 420, 122 | nsyli 160 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡))) |
422 | 421 | impcom 411 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑡)) |
423 | | difsn 4686 |
. . . . . . . . . . . . . . . 16
⊢ (¬
〈(𝑀 + 1), 0〉
∈ (1st ‘𝑡) → ((1st ‘𝑡) ∖ {〈(𝑀 + 1), 0〉}) =
(1st ‘𝑡)) |
424 | 422, 423 | 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)) |
425 | | xp1st 7746 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st ‘𝑛) ∈ ((0..^𝐾) ↑m (1...𝑀))) |
426 | | elmapfn 8475 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑛) ∈ ((0..^𝐾) ↑m (1...𝑀)) → (1st ‘𝑛) Fn (1...𝑀)) |
427 | | fnop 6446 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1st ‘𝑛) Fn (1...𝑀) ∧ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛)) → (𝑀 + 1) ∈ (1...𝑀)) |
428 | 427 | ex 416 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑛) Fn (1...𝑀) → (〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛) → (𝑀 + 1) ∈ (1...𝑀))) |
429 | 425, 426,
428 | 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))) |
430 | 429, 122 | nsyli 160 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛))) |
431 | 430 | impcom 411 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ 〈(𝑀 + 1), 0〉 ∈ (1st
‘𝑛)) |
432 | | difsn 4686 |
. . . . . . . . . . . . . . . 16
⊢ (¬
〈(𝑀 + 1), 0〉
∈ (1st ‘𝑛) → ((1st ‘𝑛) ∖ {〈(𝑀 + 1), 0〉}) =
(1st ‘𝑛)) |
433 | 431, 432 | 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)) |
434 | 424, 433 | eqeqan12d 2755 |
. . . . . . . . . . . . . 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))) |
435 | 434 | anandis 678 |
. . . . . . . . . . . . 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))) |
436 | | f1ofn 6619 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd ‘𝑡) Fn (1...𝑀)) |
437 | | fnop 6446 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘𝑡) Fn (1...𝑀) ∧ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡)) → (𝑀 + 1) ∈ (1...𝑀)) |
438 | 437 | ex 416 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑡) Fn (1...𝑀) → (〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡) → (𝑀 + 1) ∈ (1...𝑀))) |
439 | 17, 436, 438 | 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))) |
440 | 439, 122 | nsyli 160 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡))) |
441 | 440 | impcom 411 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑡)) |
442 | | difsn 4686 |
. . . . . . . . . . . . . . . 16
⊢ (¬
〈(𝑀 + 1), (𝑀 + 1)〉 ∈
(2nd ‘𝑡)
→ ((2nd ‘𝑡) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = (2nd ‘𝑡)) |
443 | 441, 442 | 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)) |
444 | | xp2nd 7747 |
. . . . . . . . . . . . . . . . . . . 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)}) |
445 | | fvex 6687 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(2nd ‘𝑛) ∈ V |
446 | | f1oeq1 6606 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓 = (2nd ‘𝑛) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀))) |
447 | 445, 446 | elab 3573 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘𝑛) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀)) |
448 | 444, 447 | sylib 221 |
. . . . . . . . . . . . . . . . . . 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)) |
449 | | f1ofn 6619 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd ‘𝑛) Fn (1...𝑀)) |
450 | | fnop 6446 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘𝑛) Fn (1...𝑀) ∧ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛)) → (𝑀 + 1) ∈ (1...𝑀)) |
451 | 450 | ex 416 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑛) Fn (1...𝑀) → (〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛) → (𝑀 + 1) ∈ (1...𝑀))) |
452 | 448, 449,
451 | 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))) |
453 | 452, 122 | nsyli 160 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛))) |
454 | 453 | impcom 411 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ 〈(𝑀 + 1), (𝑀 + 1)〉 ∈ (2nd
‘𝑛)) |
455 | | difsn 4686 |
. . . . . . . . . . . . . . . 16
⊢ (¬
〈(𝑀 + 1), (𝑀 + 1)〉 ∈
(2nd ‘𝑛)
→ ((2nd ‘𝑛) ∖ {〈(𝑀 + 1), (𝑀 + 1)〉}) = (2nd ‘𝑛)) |
456 | 454, 455 | 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)) |
457 | 443, 456 | eqeqan12d 2755 |
. . . . . . . . . . . . . 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))) |
458 | 457 | anandis 678 |
. . . . . . . . . . . . 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))) |
459 | 435, 458 | anbi12d 634 |
. . . . . . . . . . . 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)))) |
460 | | xpopth 7755 |
. . . . . . . . . . . . 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)) |
461 | 460 | adantl 485 |
. . . . . . . . . . . 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)) |
462 | 459, 461 | bitrd 282 |
. . . . . . . . . . 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)) |
463 | 416, 462 | syl5ib 247 |
. . . . . . . . . 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)) |
464 | 403, 463 | sylan2 596 |
. . . . . . . . 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)) |
465 | 464 | ralrimivva 3103 |
. . . . . . . 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)) |
466 | 465 | adantr 484 |
. . . . . . 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)) |
467 | 400, 466 | jctird 530 |
. . . . . 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)))) |
468 | | fveq2 6674 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑛 → (1st ‘𝑡) = (1st ‘𝑛)) |
469 | 468 | uneq1d 4052 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑛 → ((1st ‘𝑡) ∪ {〈(𝑀 + 1), 0〉}) =
((1st ‘𝑛)
∪ {〈(𝑀 + 1),
0〉})) |
470 | | fveq2 6674 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑛 → (2nd ‘𝑡) = (2nd ‘𝑛)) |
471 | 470 | uneq1d 4052 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑛 → ((2nd ‘𝑡) ∪ {〈(𝑀 + 1), (𝑀 + 1)〉}) = ((2nd
‘𝑛) ∪
{〈(𝑀 + 1), (𝑀 + 1)〉})) |
472 | 469, 471 | opeq12d 4769 |
. . . . . . . . 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)〉})〉) |
473 | 472 | eqeq2d 2749 |
. . . . . . . 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)〉})〉)) |
474 | 473 | reu4 3630 |
. . . . . . 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))) |
475 | 58 | rexrab 3595 |
. . . . . . . 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)〉})〉)) |
476 | 475 | anbi1i 627 |
. . . . . . 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))) |
477 | 474, 476 | bitri 278 |
. . . . . 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))) |
478 | 467, 477 | syl6ibr 255 |
. . . . 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)〉})〉)) |
479 | 478 | ralrimiva 3096 |
. . . 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)〉})〉)) |
480 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑘 → (1st ‘𝑠) = (1st ‘𝑘)) |
481 | | fveq2 6674 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑘 → (2nd ‘𝑠) = (2nd ‘𝑘)) |
482 | 481 | imaeq1d 5902 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd
‘𝑘) “
(1...𝑗))) |
483 | 482 | xpeq1d 5554 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) =
(((2nd ‘𝑘)
“ (1...𝑗)) ×
{1})) |
484 | 481 | imaeq1d 5902 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) |
485 | 484 | xpeq1d 5554 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((2nd
‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) |
486 | 483, 485 | uneq12d 4054 |
. . . . . . . . . . . 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}))) |
487 | 480, 486 | oveq12d 7188 |
. . . . . . . . . . 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})))) |
488 | 487 | uneq1d 4052 |
. . . . . . . . . 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}))) |
489 | 488 | csbeq1d 3794 |
. . . . . . . . 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) |
490 | 489 | eqeq2d 2749 |
. . . . . . . 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)) |
491 | 490 | rexbidv 3207 |
. . . . . . 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)) |
492 | 491 | ralbidv 3109 |
. . . . . 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)) |
493 | 480 | fveq1d 6676 |
. . . . . . 7
⊢ (𝑠 = 𝑘 → ((1st ‘𝑠)‘(𝑀 + 1)) = ((1st ‘𝑘)‘(𝑀 + 1))) |
494 | 493 | eqeq1d 2740 |
. . . . . 6
⊢ (𝑠 = 𝑘 → (((1st ‘𝑠)‘(𝑀 + 1)) = 0 ↔ ((1st
‘𝑘)‘(𝑀 + 1)) = 0)) |
495 | 481 | fveq1d 6676 |
. . . . . . 7
⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠)‘(𝑀 + 1)) = ((2nd ‘𝑘)‘(𝑀 + 1))) |
496 | 495 | eqeq1d 2740 |
. . . . . 6
⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) |
497 | 492, 494,
496 | 3anbi123d 1437 |
. . . . 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)))) |
498 | 497 | ralrab 3593 |
. . . 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)〉})〉)) |
499 | 479, 498 | sylibr 237 |
. . 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)〉})〉) |
500 | | eqid 2738 |
. . . 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)〉})〉) |
501 | 500 | f1ompt 6885 |
. . 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)〉})〉)) |
502 | 60, 499, 501 | sylanbrc 586 |
. 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))}) |
503 | | ovex 7203 |
. . . . 5
⊢
((0..^𝐾)
↑m (1...𝑀))
∈ V |
504 | | ovex 7203 |
. . . . . 6
⊢
((1...𝑀)
↑m (1...𝑀))
∈ V |
505 | | f1of 6618 |
. . . . . . . 8
⊢ (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑓:(1...𝑀)⟶(1...𝑀)) |
506 | 505 | ss2abi 3956 |
. . . . . . 7
⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ {𝑓 ∣ 𝑓:(1...𝑀)⟶(1...𝑀)} |
507 | 68, 68 | mapval 8449 |
. . . . . . 7
⊢
((1...𝑀)
↑m (1...𝑀))
= {𝑓 ∣ 𝑓:(1...𝑀)⟶(1...𝑀)} |
508 | 506, 507 | sseqtrri 3914 |
. . . . . 6
⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ ((1...𝑀) ↑m (1...𝑀)) |
509 | 504, 508 | ssexi 5190 |
. . . . 5
⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ∈ V |
510 | 503, 509 | xpex 7494 |
. . . 4
⊢
(((0..^𝐾)
↑m (1...𝑀))
× {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∈ V |
511 | 510 | rabex 5200 |
. . 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 |
512 | 511 | f1oen 8576 |
. 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))}) |
513 | 502, 512 | 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))}) |