Step | Hyp | Ref
| Expression |
1 | | poimir.0 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) |
2 | 1 | nnnn0d 12293 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
3 | 1 | nnred 11988 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℝ) |
4 | 3 | leidd 11541 |
. . . . 5
⊢ (𝜑 → 𝑁 ≤ 𝑁) |
5 | 2, 2, 4 | 3jca 1127 |
. . . 4
⊢ (𝜑 → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0
∧ 𝑁 ≤ 𝑁)) |
6 | | oveq2 7279 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → (1...𝑘) = (1...0)) |
7 | | fz10 13276 |
. . . . . . . . . . . . . . . 16
⊢ (1...0) =
∅ |
8 | 6, 7 | eqtrdi 2796 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → (1...𝑘) = ∅) |
9 | 8 | oveq2d 7287 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m
∅)) |
10 | | fzofi 13692 |
. . . . . . . . . . . . . . . 16
⊢
(0..^𝐾) ∈
Fin |
11 | | map0e 8653 |
. . . . . . . . . . . . . . . 16
⊢
((0..^𝐾) ∈ Fin
→ ((0..^𝐾)
↑m ∅) = 1o) |
12 | 10, 11 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
((0..^𝐾)
↑m ∅) = 1o |
13 | | df1o2 8295 |
. . . . . . . . . . . . . . 15
⊢
1o = {∅} |
14 | 12, 13 | eqtri 2768 |
. . . . . . . . . . . . . 14
⊢
((0..^𝐾)
↑m ∅) = {∅} |
15 | 9, 14 | eqtrdi 2796 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → ((0..^𝐾) ↑m (1...𝑘)) = {∅}) |
16 | | eqidd 2741 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → 𝑓 = 𝑓) |
17 | 16, 8, 8 | f1oeq123d 6708 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:∅–1-1-onto→∅)) |
18 | | eqid 2740 |
. . . . . . . . . . . . . . . . 17
⊢ ∅ =
∅ |
19 | | f1o00 6748 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:∅–1-1-onto→∅ ↔ (𝑓 = ∅ ∧ ∅ =
∅)) |
20 | 18, 19 | mpbiran2 707 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:∅–1-1-onto→∅ ↔ 𝑓 = ∅) |
21 | 17, 20 | bitrdi 287 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓 = ∅)) |
22 | 21 | abbidv 2809 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓 = ∅}) |
23 | | df-sn 4568 |
. . . . . . . . . . . . . 14
⊢ {∅}
= {𝑓 ∣ 𝑓 = ∅} |
24 | 22, 23 | eqtr4di 2798 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {∅}) |
25 | 15, 24 | xpeq12d 5621 |
. . . . . . . . . . . 12
⊢ (𝑘 = 0 → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = ({∅} ×
{∅})) |
26 | | 0ex 5235 |
. . . . . . . . . . . . 13
⊢ ∅
∈ V |
27 | 26, 26 | xpsn 7010 |
. . . . . . . . . . . 12
⊢
({∅} × {∅}) = {〈∅,
∅〉} |
28 | 25, 27 | eqtr2di 2797 |
. . . . . . . . . . 11
⊢ (𝑘 = 0 → {〈∅,
∅〉} = (((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)})) |
29 | | elsni 4584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {〈∅,
∅〉} → 𝑠 =
〈∅, ∅〉) |
30 | 26, 26 | op1std 7834 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = 〈∅, ∅〉
→ (1st ‘𝑠) = ∅) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {〈∅,
∅〉} → (1st ‘𝑠) = ∅) |
32 | 31 | oveq1d 7286 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {〈∅,
∅〉} → ((1st ‘𝑠) ∘f + ∅) = (∅
∘f + ∅)) |
33 | | f0 6653 |
. . . . . . . . . . . . . . . . . . . 20
⊢
∅:∅⟶∅ |
34 | | ffn 6598 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∅:∅⟶∅ → ∅ Fn
∅) |
35 | 33, 34 | mp1i 13 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {〈∅,
∅〉} → ∅ Fn ∅) |
36 | 26 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {〈∅,
∅〉} → ∅ ∈ V) |
37 | | inidm 4158 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∅
∩ ∅) = ∅ |
38 | | 0fv 6810 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∅‘𝑛) =
∅ |
39 | 38 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ {〈∅,
∅〉} ∧ 𝑛
∈ ∅) → (∅‘𝑛) = ∅) |
40 | 35, 35, 36, 36, 37, 39, 39 | offval 7536 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {〈∅,
∅〉} → (∅ ∘f + ∅) = (𝑛 ∈ ∅ ↦ (∅
+ ∅))) |
41 | | mpt0 6573 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ∅ ↦ (∅
+ ∅)) = ∅ |
42 | 40, 41 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {〈∅,
∅〉} → (∅ ∘f + ∅) =
∅) |
43 | 32, 42 | eqtrd 2780 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈ {〈∅,
∅〉} → ((1st ‘𝑠) ∘f + ∅) =
∅) |
44 | 43 | uneq1d 4101 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ {〈∅,
∅〉} → (((1st ‘𝑠) ∘f + ∅) ∪
((1...𝑁) × {0})) =
(∅ ∪ ((1...𝑁)
× {0}))) |
45 | | uncom 4092 |
. . . . . . . . . . . . . . . 16
⊢ (∅
∪ ((1...𝑁) ×
{0})) = (((1...𝑁) ×
{0}) ∪ ∅) |
46 | | un0 4330 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑁) ×
{0}) ∪ ∅) = ((1...𝑁) × {0}) |
47 | 45, 46 | eqtri 2768 |
. . . . . . . . . . . . . . 15
⊢ (∅
∪ ((1...𝑁) ×
{0})) = ((1...𝑁) ×
{0}) |
48 | 44, 47 | eqtr2di 2797 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ {〈∅,
∅〉} → ((1...𝑁) × {0}) = (((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0}))) |
49 | 48 | csbeq1d 3841 |
. . . . . . . . . . . . 13
⊢ (𝑠 ∈ {〈∅,
∅〉} → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵) |
50 | 49 | eqeq2d 2751 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ {〈∅,
∅〉} → (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
51 | | oveq2 7279 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → (0...𝑘) = (0...0)) |
52 | | 0z 12330 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℤ |
53 | | fzsn 13297 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
ℤ → (0...0) = {0}) |
54 | 52, 53 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (0...0) =
{0} |
55 | 51, 54 | eqtrdi 2796 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (0...𝑘) = {0}) |
56 | | oveq2 7279 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 0 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...0)) |
57 | 56 | imaeq2d 5968 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 0 → ((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...0))) |
58 | 57 | xpeq1d 5619 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 0 → (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) ×
{0})) |
59 | 58 | uneq2d 4102 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0})) =
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) ×
{0}))) |
60 | 59 | oveq2d 7287 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → ((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) ×
{0})))) |
61 | | oveq1 7278 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1)) |
62 | | 0p1e1 12095 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0 + 1) =
1 |
63 | 61, 62 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 0 → (𝑘 + 1) = 1) |
64 | 63 | oveq1d 7286 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((𝑘 + 1)...𝑁) = (1...𝑁)) |
65 | 64 | xpeq1d 5619 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (((𝑘 + 1)...𝑁) × {0}) = ((1...𝑁) × {0})) |
66 | 60, 65 | uneq12d 4103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) ×
{0}))) |
67 | 66 | csbeq1d 3841 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 →
⦋(((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵) |
68 | 67 | eqeq2d 2751 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → (𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵)) |
69 | 55, 68 | rexeqbidv 3336 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ {0}𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵)) |
70 | | c0ex 10970 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
V |
71 | | oveq2 7279 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 0 → (1...𝑗) = (1...0)) |
72 | 71, 7 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = 0 → (1...𝑗) = ∅) |
73 | 72 | imaeq2d 5968 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 0 → ((2nd
‘𝑠) “
(1...𝑗)) = ((2nd
‘𝑠) “
∅)) |
74 | | ima0 5984 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((2nd ‘𝑠) “ ∅) = ∅ |
75 | 73, 74 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 0 → ((2nd
‘𝑠) “
(1...𝑗)) =
∅) |
76 | 75 | xpeq1d 5619 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 0 → (((2nd
‘𝑠) “
(1...𝑗)) × {1}) =
(∅ × {1})) |
77 | | 0xp 5685 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (∅
× {1}) = ∅ |
78 | 76, 77 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 0 → (((2nd
‘𝑠) “
(1...𝑗)) × {1}) =
∅) |
79 | | oveq1 7278 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1)) |
80 | 79, 62 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 0 → (𝑗 + 1) = 1) |
81 | 80 | oveq1d 7286 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = (1...0)) |
82 | 81, 7 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = ∅) |
83 | 82 | imaeq2d 5968 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 0 → ((2nd
‘𝑠) “ ((𝑗 + 1)...0)) = ((2nd
‘𝑠) “
∅)) |
84 | 83, 74 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 0 → ((2nd
‘𝑠) “ ((𝑗 + 1)...0)) =
∅) |
85 | 84 | xpeq1d 5619 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 0 → (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}) =
(∅ × {0})) |
86 | | 0xp 5685 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (∅
× {0}) = ∅ |
87 | 85, 86 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 0 → (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}) =
∅) |
88 | 78, 87 | uneq12d 4103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 0 → ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...0))
× {0})) = (∅ ∪ ∅)) |
89 | | un0 4330 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∅
∪ ∅) = ∅ |
90 | 88, 89 | eqtrdi 2796 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 0 → ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...0))
× {0})) = ∅) |
91 | 90 | oveq2d 7287 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 0 → ((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) =
((1st ‘𝑠)
∘f + ∅)) |
92 | 91 | uneq1d 4101 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 0 → (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) =
(((1st ‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0}))) |
93 | 92 | csbeq1d 3841 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 0 →
⦋(((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...0))
× {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵) |
94 | 93 | eqeq2d 2751 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 0 → (𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
95 | 70, 94 | rexsn 4624 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑗 ∈
{0}𝑖 =
⦋(((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...0))
× {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵) |
96 | 69, 95 | bitrdi 287 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
97 | 55, 96 | raleqbidv 3335 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 0 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ {0}𝑖 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
98 | | eqeq1 2744 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 0 → (𝑖 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st
‘𝑠)
∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)) |
99 | 70, 98 | ralsn 4623 |
. . . . . . . . . . . . 13
⊢
(∀𝑖 ∈
{0}𝑖 =
⦋(((1st ‘𝑠) ∘f + ∅) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵 ↔ 0 =
⦋(((1st ‘𝑠) ∘f + ∅) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵) |
100 | 97, 99 | bitr2di 288 |
. . . . . . . . . . . 12
⊢ (𝑘 = 0 → (0 =
⦋(((1st ‘𝑠) ∘f + ∅) ∪
((1...𝑁) × {0})) /
𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
101 | 50, 100 | sylan9bbr 511 |
. . . . . . . . . . 11
⊢ ((𝑘 = 0 ∧ 𝑠 ∈ {〈∅, ∅〉})
→ (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
102 | 28, 101 | rabeqbidva 3420 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → {𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) |
103 | 102 | eqcomd 2746 |
. . . . . . . . 9
⊢ (𝑘 = 0 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}) |
104 | 103 | fveq2d 6775 |
. . . . . . . 8
⊢ (𝑘 = 0 →
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})) |
105 | 104 | breq2d 5091 |
. . . . . . 7
⊢ (𝑘 = 0 → (2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ {〈∅,
∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))) |
106 | 105 | notbid 318 |
. . . . . 6
⊢ (𝑘 = 0 → (¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥
(♯‘{𝑠 ∈
{〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))) |
107 | 106 | imbi2d 341 |
. . . . 5
⊢ (𝑘 = 0 → ((𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
{〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))) |
108 | | oveq2 7279 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → (1...𝑘) = (1...𝑚)) |
109 | 108 | oveq2d 7287 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...𝑚))) |
110 | | eqidd 2741 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → 𝑓 = 𝑓) |
111 | 110, 108,
108 | f1oeq123d 6708 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚))) |
112 | 111 | abbidv 2809 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) |
113 | 109, 112 | xpeq12d 5621 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑚 → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)})) |
114 | | oveq2 7279 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → (0...𝑘) = (0...𝑚)) |
115 | | oveq2 7279 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑚 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑚)) |
116 | 115 | imaeq2d 5968 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑚 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚))) |
117 | 116 | xpeq1d 5619 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0})) |
118 | 117 | uneq2d 4102 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0})) =
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) |
119 | 118 | oveq2d 7287 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑚 → ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0})))) |
120 | | oveq1 7278 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1)) |
121 | 120 | oveq1d 7286 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → ((𝑘 + 1)...𝑁) = ((𝑚 + 1)...𝑁)) |
122 | 121 | xpeq1d 5619 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑚 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑚 + 1)...𝑁) × {0})) |
123 | 119, 122 | uneq12d 4103 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑚 → (((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0}))) |
124 | 123 | csbeq1d 3841 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
125 | 124 | eqeq2d 2751 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → (𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
126 | 114, 125 | rexeqbidv 3336 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
127 | 114, 126 | raleqbidv 3335 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑚 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀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)) |
128 | 113, 127 | rabeqbidv 3419 |
. . . . . . . . 9
⊢ (𝑘 = 𝑚 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {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}) |
129 | 128 | fveq2d 6775 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{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})) |
130 | 129 | breq2d 5091 |
. . . . . . 7
⊢ (𝑘 = 𝑚 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{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}))) |
131 | 130 | notbid 318 |
. . . . . 6
⊢ (𝑘 = 𝑚 → (¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥
(♯‘{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}))) |
132 | 131 | imbi2d 341 |
. . . . 5
⊢ (𝑘 = 𝑚 → ((𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥
(♯‘{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})))) |
133 | | oveq2 7279 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑚 + 1) → (1...𝑘) = (1...(𝑚 + 1))) |
134 | 133 | oveq2d 7287 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 + 1) → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...(𝑚 + 1)))) |
135 | | eqidd 2741 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝑚 + 1) → 𝑓 = 𝑓) |
136 | 135, 133,
133 | f1oeq123d 6708 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑚 + 1) → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)))) |
137 | 136 | abbidv 2809 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 + 1) → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) |
138 | 134, 137 | xpeq12d 5621 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑚 + 1) → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))})) |
139 | | oveq2 7279 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 + 1) → (0...𝑘) = (0...(𝑚 + 1))) |
140 | | oveq2 7279 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (𝑚 + 1) → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...(𝑚 + 1))) |
141 | 140 | imaeq2d 5968 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = (𝑚 + 1) → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1)))) |
142 | 141 | xpeq1d 5619 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (𝑚 + 1) → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})) |
143 | 142 | uneq2d 4102 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑚 + 1) → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0})) =
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) |
144 | 143 | oveq2d 7287 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑚 + 1) → ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})))) |
145 | | oveq1 7278 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = (𝑚 + 1) → (𝑘 + 1) = ((𝑚 + 1) + 1)) |
146 | 145 | oveq1d 7286 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (𝑚 + 1) → ((𝑘 + 1)...𝑁) = (((𝑚 + 1) + 1)...𝑁)) |
147 | 146 | xpeq1d 5619 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (𝑚 + 1) → (((𝑘 + 1)...𝑁) × {0}) = ((((𝑚 + 1) + 1)...𝑁) × {0})) |
148 | 144, 147 | uneq12d 4103 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (𝑚 + 1) → (((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))) |
149 | 148 | csbeq1d 3841 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝑚 + 1) → ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
150 | 149 | eqeq2d 2751 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑚 + 1) → (𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
151 | 139, 150 | rexeqbidv 3336 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑚 + 1) → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...(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)) |
152 | 139, 151 | raleqbidv 3335 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑚 + 1) → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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)) |
153 | 138, 152 | rabeqbidv 3419 |
. . . . . . . . 9
⊢ (𝑘 = (𝑚 + 1) → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 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 + 1))∃𝑗 ∈ (0...(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}) |
154 | 153 | fveq2d 6775 |
. . . . . . . 8
⊢ (𝑘 = (𝑚 + 1) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 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 + 1))∃𝑗 ∈ (0...(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})) |
155 | 154 | breq2d 5091 |
. . . . . . 7
⊢ (𝑘 = (𝑚 + 1) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}))) |
156 | 155 | notbid 318 |
. . . . . 6
⊢ (𝑘 = (𝑚 + 1) → (¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}))) |
157 | 156 | imbi2d 341 |
. . . . 5
⊢ (𝑘 = (𝑚 + 1) → ((𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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})))) |
158 | | oveq2 7279 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑁 → (1...𝑘) = (1...𝑁)) |
159 | 158 | oveq2d 7287 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑁 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...𝑁))) |
160 | | eqidd 2741 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑁 → 𝑓 = 𝑓) |
161 | 160, 158,
158 | f1oeq123d 6708 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑁 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))) |
162 | 161 | abbidv 2809 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑁 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
163 | 159, 162 | xpeq12d 5621 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑁 → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
164 | | oveq2 7279 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑁 → (0...𝑘) = (0...𝑁)) |
165 | | oveq2 7279 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑁 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑁)) |
166 | 165 | imaeq2d 5968 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑁 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁))) |
167 | 166 | xpeq1d 5619 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑁 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) |
168 | 167 | uneq2d 4102 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑁 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑘)) × {0})) =
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) |
169 | 168 | oveq2d 7287 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑁 → ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))) |
170 | | oveq1 7278 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1)) |
171 | 170 | oveq1d 7286 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑁 → ((𝑘 + 1)...𝑁) = ((𝑁 + 1)...𝑁)) |
172 | 171 | xpeq1d 5619 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑁 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑁 + 1)...𝑁) × {0})) |
173 | 169, 172 | uneq12d 4103 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑁 → (((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0}))) |
174 | 173 | csbeq1d 3841 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑁 → ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
175 | 174 | eqeq2d 2751 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑁 → (𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
176 | 164, 175 | rexeqbidv 3336 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑁 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
177 | 164, 176 | raleqbidv 3335 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑁 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
178 | 163, 177 | rabeqbidv 3419 |
. . . . . . . . 9
⊢ (𝑘 = 𝑁 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) |
179 | 178 | fveq2d 6775 |
. . . . . . . 8
⊢ (𝑘 = 𝑁 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) |
180 | 179 | breq2d 5091 |
. . . . . . 7
⊢ (𝑘 = 𝑁 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
181 | 180 | notbid 318 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
182 | 181 | imbi2d 341 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑘))
× {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))) |
183 | | n2dvds1 16075 |
. . . . . . 7
⊢ ¬ 2
∥ 1 |
184 | | opex 5383 |
. . . . . . . . . 10
⊢
〈∅, ∅〉 ∈ V |
185 | | hashsng 14082 |
. . . . . . . . . 10
⊢
(〈∅, ∅〉 ∈ V →
(♯‘{〈∅, ∅〉}) = 1) |
186 | 184, 185 | ax-mp 5 |
. . . . . . . . 9
⊢
(♯‘{〈∅, ∅〉}) = 1 |
187 | | nnuz 12620 |
. . . . . . . . . . . . . . . . 17
⊢ ℕ =
(ℤ≥‘1) |
188 | 1, 187 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
189 | | eluzfz1 13262 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘1) → 1 ∈ (1...𝑁)) |
190 | 188, 189 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ∈ (1...𝑁)) |
191 | | poimirlem28.5 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐾 ∈ ℕ) |
192 | 191 | nnnn0d 12293 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐾 ∈
ℕ0) |
193 | | 0elfz 13352 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ∈ ℕ0
→ 0 ∈ (0...𝐾)) |
194 | | fconst6g 6661 |
. . . . . . . . . . . . . . . 16
⊢ (0 ∈
(0...𝐾) → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)) |
195 | 192, 193,
194 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)) |
196 | 70 | fvconst2 7076 |
. . . . . . . . . . . . . . . 16
⊢ (1 ∈
(1...𝑁) → (((1...𝑁) × {0})‘1) =
0) |
197 | 190, 196 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((1...𝑁) × {0})‘1) =
0) |
198 | 190, 195,
197 | 3jca 1127 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) =
0)) |
199 | | nfv 1921 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑝(𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) =
0)) |
200 | | nfcsb1v 3862 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 |
201 | 200 | nfeq1 2924 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0 |
202 | 199, 201 | nfim 1903 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑝((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) →
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵 = 0) |
203 | | ovex 7304 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑁) ∈
V |
204 | | snex 5358 |
. . . . . . . . . . . . . . . 16
⊢ {0}
∈ V |
205 | 203, 204 | xpex 7597 |
. . . . . . . . . . . . . . 15
⊢
((1...𝑁) ×
{0}) ∈ V |
206 | | feq1 6579 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))) |
207 | | fveq1 6770 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝‘1) = (((1...𝑁) × {0})‘1)) |
208 | 207 | eqeq1d 2742 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝑝‘1) = 0 ↔ (((1...𝑁) × {0})‘1) =
0)) |
209 | 206, 208 | 3anbi23d 1438 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 = ((1...𝑁) × {0}) → ((1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0) ↔ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) =
0))) |
210 | 209 | anbi2d 629 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) =
0)))) |
211 | | csbeq1a 3851 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 = ((1...𝑁) × {0}) → 𝐵 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵) |
212 | 211 | eqeq1d 2742 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 = ((1...𝑁) × {0}) → (𝐵 = 0 ↔ ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)) |
213 | 210, 212 | imbi12d 345 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = ((1...𝑁) × {0}) → (((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) →
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵 = 0))) |
214 | | 1ex 10972 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
V |
215 | | eleq1 2828 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 1 → (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁))) |
216 | | fveqeq2 6780 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 1 → ((𝑝‘𝑛) = 0 ↔ (𝑝‘1) = 0)) |
217 | 215, 216 | 3anbi13d 1437 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 1 → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0))) |
218 | 217 | anbi2d 629 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 1 → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)))) |
219 | | breq2 5083 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 1 → (𝐵 < 𝑛 ↔ 𝐵 < 1)) |
220 | 218, 219 | imbi12d 345 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 1 → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1))) |
221 | | poimirlem28.3 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) |
222 | 214, 220,
221 | vtocl 3497 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1) |
223 | | poimirlem28.2 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) |
224 | | elfznn0 13348 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈
ℕ0) |
225 | | nn0lt10b 12382 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ ℕ0
→ (𝐵 < 1 ↔
𝐵 = 0)) |
226 | 223, 224,
225 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝐵 < 1 ↔ 𝐵 = 0)) |
227 | 226 | 3ad2antr2 1188 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → (𝐵 < 1 ↔ 𝐵 = 0)) |
228 | 222, 227 | mpbid 231 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0) |
229 | 202, 205,
213, 228 | vtoclf 3496 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) →
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵 = 0) |
230 | 198, 229 | mpdan 684 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0) |
231 | 230 | eqcomd 2746 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 =
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵) |
232 | 231 | ralrimivw 3111 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑠 ∈ {〈∅, ∅〉}0 =
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵) |
233 | | rabid2 3313 |
. . . . . . . . . . 11
⊢
({〈∅, ∅〉} = {𝑠 ∈ {〈∅, ∅〉}
∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} ↔ ∀𝑠 ∈ {〈∅, ∅〉}0 =
⦋((1...𝑁)
× {0}) / 𝑝⦌𝐵) |
234 | 232, 233 | sylibr 233 |
. . . . . . . . . 10
⊢ (𝜑 → {〈∅,
∅〉} = {𝑠 ∈
{〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}) |
235 | 234 | fveq2d 6775 |
. . . . . . . . 9
⊢ (𝜑 →
(♯‘{〈∅, ∅〉}) = (♯‘{𝑠 ∈ {〈∅,
∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})) |
236 | 186, 235 | eqtr3id 2794 |
. . . . . . . 8
⊢ (𝜑 → 1 = (♯‘{𝑠 ∈ {〈∅,
∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})) |
237 | 236 | breq2d 5091 |
. . . . . . 7
⊢ (𝜑 → (2 ∥ 1 ↔ 2
∥ (♯‘{𝑠
∈ {〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))) |
238 | 183, 237 | mtbii 326 |
. . . . . 6
⊢ (𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
{〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})) |
239 | 238 | a1i 11 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ (𝜑 → ¬ 2
∥ (♯‘{𝑠
∈ {〈∅, ∅〉} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))) |
240 | | 2z 12352 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℤ |
241 | | fzfi 13690 |
. . . . . . . . . . . . . . . . 17
⊢
(1...(𝑚 + 1)) ∈
Fin |
242 | | mapfi 9093 |
. . . . . . . . . . . . . . . . 17
⊢
(((0..^𝐾) ∈ Fin
∧ (1...(𝑚 + 1)) ∈
Fin) → ((0..^𝐾)
↑m (1...(𝑚
+ 1))) ∈ Fin) |
243 | 10, 241, 242 | mp2an 689 |
. . . . . . . . . . . . . . . 16
⊢
((0..^𝐾)
↑m (1...(𝑚
+ 1))) ∈ Fin |
244 | | ovex 7304 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1...(𝑚 + 1)) ∈
V |
245 | 244, 244 | mapval 8610 |
. . . . . . . . . . . . . . . . . 18
⊢
((1...(𝑚 + 1))
↑m (1...(𝑚
+ 1))) = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} |
246 | | mapfi 9093 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...(𝑚 + 1))
∈ Fin ∧ (1...(𝑚 +
1)) ∈ Fin) → ((1...(𝑚 + 1)) ↑m (1...(𝑚 + 1))) ∈
Fin) |
247 | 241, 241,
246 | mp2an 689 |
. . . . . . . . . . . . . . . . . 18
⊢
((1...(𝑚 + 1))
↑m (1...(𝑚
+ 1))) ∈ Fin |
248 | 245, 247 | eqeltrri 2838 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin |
249 | | f1of 6714 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)) → 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))) |
250 | 249 | ss2abi 4005 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} |
251 | | ssfi 8938 |
. . . . . . . . . . . . . . . . 17
⊢ (({𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}) → {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin) |
252 | 248, 250,
251 | mp2an 689 |
. . . . . . . . . . . . . . . 16
⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin |
253 | | xpfi 9063 |
. . . . . . . . . . . . . . . 16
⊢
((((0..^𝐾)
↑m (1...(𝑚
+ 1))) ∈ Fin ∧ {𝑓
∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin) → (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin) |
254 | 243, 252,
253 | mp2an 689 |
. . . . . . . . . . . . . . 15
⊢
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin |
255 | | rabfi 9022 |
. . . . . . . . . . . . . . 15
⊢
((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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} ∈ Fin) |
256 | | hashcl 14069 |
. . . . . . . . . . . . . . 15
⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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} ∈ Fin → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ∈
ℕ0) |
257 | 254, 255,
256 | mp2b 10 |
. . . . . . . . . . . . . 14
⊢
(♯‘{𝑠
∈ (((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ∈
ℕ0 |
258 | 257 | nn0zi 12345 |
. . . . . . . . . . . . 13
⊢
(♯‘{𝑠
∈ (((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ∈ ℤ |
259 | | rabfi 9022 |
. . . . . . . . . . . . . . 15
⊢
((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))} ∈ Fin) |
260 | | hashcl 14069 |
. . . . . . . . . . . . . . 15
⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))} ∈ Fin →
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ∈
ℕ0) |
261 | 254, 259,
260 | mp2b 10 |
. . . . . . . . . . . . . 14
⊢
(♯‘{𝑠
∈ (((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ∈
ℕ0 |
262 | 261 | nn0zi 12345 |
. . . . . . . . . . . . 13
⊢
(♯‘{𝑠
∈ (((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ∈ ℤ |
263 | 240, 258,
262 | 3pm3.2i 1338 |
. . . . . . . . . . . 12
⊢ (2 ∈
ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ∈ ℤ ∧
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ∈ ℤ) |
264 | | nn0p1nn 12272 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℕ) |
265 | 264 | ad2antrl 725 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ∈ ℕ) |
266 | | uneq1 4095 |
. . . . . . . . . . . . . . . 16
⊢ (𝑞 = ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))) |
267 | 266 | csbeq1d 3841 |
. . . . . . . . . . . . . . 15
⊢ (𝑞 = ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) →
⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
268 | 70 | fconst 6658 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0} |
269 | 268 | jctr 525 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0})) |
270 | 264 | nnred 11988 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℝ) |
271 | 270 | ltp1d 11905 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) < ((𝑚 + 1) + 1)) |
272 | | fzdisj 13282 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑚 + 1) < ((𝑚 + 1) + 1) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) |
273 | 271, 272 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈ ℕ0
→ ((1...(𝑚 + 1)) ∩
(((𝑚 + 1) + 1)...𝑁)) = ∅) |
274 | | fun 6634 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}) ∧ ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0})) |
275 | 269, 273,
274 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 ∈ ℕ0
∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0})) |
276 | 275 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0})) |
277 | 276 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0})) |
278 | 264 | peano2nnd 11990 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 ∈ ℕ0
→ ((𝑚 + 1) + 1) ∈
ℕ) |
279 | 278, 187 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ ((𝑚 + 1) + 1) ∈
(ℤ≥‘1)) |
280 | 279 | ad2antrl 725 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑚 + 1) + 1) ∈
(ℤ≥‘1)) |
281 | | nn0z 12343 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℤ) |
282 | 1 | nnzd 12424 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → 𝑁 ∈ ℤ) |
283 | | zltp1le 12370 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁)) |
284 | 281, 282,
283 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁)) |
285 | 284 | biimpa 477 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → (𝑚 + 1) ≤ 𝑁) |
286 | 285 | anasss 467 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ≤ 𝑁) |
287 | 281 | peano2zd 12428 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℤ) |
288 | 287 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈ ℤ) |
289 | | eluz 12595 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁)) |
290 | 288, 282,
289 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁)) |
291 | 286, 290 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑚 + 1))) |
292 | | fzsplit2 13280 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑚 + 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑚 + 1))) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))) |
293 | 280, 291,
292 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))) |
294 | 293 | eqcomd 2746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)) = (1...𝑁)) |
295 | 192, 193 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 0 ∈ (0...𝐾)) |
296 | 295 | snssd 4748 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → {0} ⊆ (0...𝐾)) |
297 | | ssequn2 4122 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({0}
⊆ (0...𝐾) ↔
((0...𝐾) ∪ {0}) =
(0...𝐾)) |
298 | 296, 297 | sylib 217 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((0...𝐾) ∪ {0}) = (0...𝐾)) |
299 | 298 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((0...𝐾) ∪ {0}) = (0...𝐾)) |
300 | 294, 299 | feq23d 6593 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))) |
301 | 300 | adantrr 714 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))) |
302 | 277, 301 | mpbid 231 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
303 | | nfv 1921 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑝(𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
304 | | nfcsb1v 3862 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 |
305 | 304 | nfel1 2925 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) |
306 | 303, 305 | nfim 1903 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑝((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
307 | | vex 3435 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑞 ∈ V |
308 | | ovex 7304 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑚 + 1) + 1)...𝑁) ∈ V |
309 | 308, 204 | xpex 7597 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) ∈ V |
310 | 307, 309 | unex 7590 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) ∈ V |
311 | | feq1 6579 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))) |
312 | 311 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) ↔ (𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))) |
313 | | csbeq1a 3851 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → 𝐵 = ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵) |
314 | 313 | eleq1d 2825 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ∈ (0...𝑁) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))) |
315 | 312, 314 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) ↔ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)))) |
316 | 306, 310,
315, 223 | vtoclf 3496 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
317 | 302, 316 | syldan 591 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
318 | 317 | anassrs 468 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
319 | | elfznn0 13348 |
. . . . . . . . . . . . . . . . 17
⊢
(⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈
ℕ0) |
320 | 318, 319 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈
ℕ0) |
321 | 264 | nnnn0d 12293 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ℕ0
→ (𝑚 + 1) ∈
ℕ0) |
322 | 321 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈
ℕ0) |
323 | 322 | ad2antlr 724 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑚 + 1) ∈
ℕ0) |
324 | | leloe 11062 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑚 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
325 | 270, 3, 324 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
326 | 284, 325 | bitrd 278 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
327 | 326 | biimpd 228 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 → ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
328 | 327 | imdistani 569 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
329 | 328 | anasss 467 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁))) |
330 | | simplll 772 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝜑) |
331 | 278 | nnge1d 12021 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 ∈ ℕ0
→ 1 ≤ ((𝑚 + 1) +
1)) |
332 | 331 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 1 ≤ ((𝑚 + 1) + 1)) |
333 | | zltp1le 12370 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
334 | 287, 282,
333 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
335 | 334 | biimpa 477 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ≤ 𝑁) |
336 | 287 | peano2zd 12428 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ ((𝑚 + 1) + 1) ∈
ℤ) |
337 | | 1z 12350 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 1 ∈
ℤ |
338 | | elfz 13244 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧
1 ∈ ℤ ∧ 𝑁
∈ ℤ) → (((𝑚
+ 1) + 1) ∈ (1...𝑁)
↔ (1 ≤ ((𝑚 + 1) +
1) ∧ ((𝑚 + 1) + 1) ≤
𝑁))) |
339 | 337, 338 | mp3an2 1448 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧
𝑁 ∈ ℤ) →
(((𝑚 + 1) + 1) ∈
(1...𝑁) ↔ (1 ≤
((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁))) |
340 | 336, 282,
339 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁))) |
341 | 340 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁))) |
342 | 332, 335,
341 | mpbir2and 710 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁)) |
343 | 342 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁)) |
344 | | nn0re 12242 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℝ) |
345 | 344 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 ∈ ℝ) |
346 | 270 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℝ) |
347 | 3 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ ℝ) |
348 | 344 | ltp1d 11905 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑚 ∈ ℕ0
→ 𝑚 < (𝑚 + 1)) |
349 | 348 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < (𝑚 + 1)) |
350 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) < 𝑁) |
351 | 345, 346,
347, 349, 350 | lttrd 11136 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁) |
352 | 351 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁) |
353 | | anass 469 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ↔ (𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁))) |
354 | 302 | anassrs 468 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
355 | 353, 354 | sylanb 581 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
356 | 355 | an32s 649 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
357 | 352, 356 | syldan 591 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
358 | | ffn 6598 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → 𝑞 Fn (1...(𝑚 + 1))) |
359 | 358 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑞 Fn (1...(𝑚 + 1))) |
360 | 273 | ad3antlr 728 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) |
361 | | eluz 12595 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧
𝑁 ∈ ℤ) →
(𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
362 | 336, 282,
361 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
363 | 362 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁)) |
364 | 335, 363 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1))) |
365 | | eluzfz1 13262 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈
(ℤ≥‘((𝑚 + 1) + 1)) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁)) |
366 | 364, 365 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁)) |
367 | 366 | adantlr 712 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁)) |
368 | | fnconstg 6660 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (0 ∈
V → ((((𝑚 + 1) +
1)...𝑁) × {0}) Fn
(((𝑚 + 1) + 1)...𝑁)) |
369 | 70, 368 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) |
370 | | fvun2 6857 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1))) |
371 | 369, 370 | mp3an2 1448 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1))) |
372 | 359, 360,
367, 371 | syl12anc 834 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1))) |
373 | 70 | fvconst2 7076 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0) |
374 | 367, 373 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0) |
375 | 372, 374 | eqtrd 2780 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0) |
376 | | nfv 1921 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑝(𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) |
377 | | nfcv 2909 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑝
< |
378 | | nfcv 2909 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑝((𝑚 + 1) + 1) |
379 | 304, 377,
378 | nfbr 5126 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1) |
380 | 376, 379 | nfim 1903 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑝((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) →
⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)) |
381 | | fveq1 6770 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘((𝑚 + 1) + 1)) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1))) |
382 | 381 | eqeq1d 2742 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘((𝑚 + 1) + 1)) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) |
383 | 311, 382 | 3anbi23d 1438 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))) |
384 | 383 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)))) |
385 | 313 | breq1d 5089 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < ((𝑚 + 1) + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))) |
386 | 384, 385 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) →
⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)))) |
387 | | ovex 7304 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 + 1) + 1) ∈
V |
388 | | eleq1 2828 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝑛 ∈ (1...𝑁) ↔ ((𝑚 + 1) + 1) ∈ (1...𝑁))) |
389 | | fveqeq2 6780 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑝‘𝑛) = 0 ↔ (𝑝‘((𝑚 + 1) + 1)) = 0)) |
390 | 388, 389 | 3anbi13d 1437 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0))) |
391 | 390 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)))) |
392 | | breq2 5083 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝐵 < 𝑛 ↔ 𝐵 < ((𝑚 + 1) + 1))) |
393 | 391, 392 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = ((𝑚 + 1) + 1) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)))) |
394 | 387, 393,
221 | vtocl 3497 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)) |
395 | 380, 310,
386, 394 | vtoclf 3496 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) →
⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)) |
396 | 330, 343,
357, 375, 395 | syl13anc 1371 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)) |
397 | 353, 318 | sylanb 581 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
398 | 397 | an32s 649 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)) |
399 | 398 | elfzelzd 13256 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ) |
400 | 352, 399 | syldan 591 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ) |
401 | 287 | ad3antlr 728 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℤ) |
402 | | zleltp1 12371 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ) →
(⦋(𝑞 ∪
((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))) |
403 | 400, 401,
402 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))) |
404 | 396, 403 | mpbird 256 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
405 | 348 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → 𝑚 < (𝑚 + 1)) |
406 | | breq2 5083 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 + 1) = 𝑁 → (𝑚 < (𝑚 + 1) ↔ 𝑚 < 𝑁)) |
407 | 406 | biimpac 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 < (𝑚 + 1) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁) |
408 | 405, 407 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁) |
409 | | elfzle2 13259 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁) |
410 | 398, 409 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁) |
411 | 408, 410 | syldan 591 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁) |
412 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑚 + 1) = 𝑁) |
413 | 411, 412 | breqtrrd 5107 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
414 | 404, 413 | jaodan 955 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
415 | 414 | an32s 649 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
416 | 329, 415 | sylan 580 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)) |
417 | | elfz2nn0 13346 |
. . . . . . . . . . . . . . . 16
⊢
(⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ∈ (0...(𝑚 + 1)) ↔ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0 ∧ (𝑚 + 1) ∈ ℕ0
∧ ⦋(𝑞
∪ ((((𝑚 + 1) +
1)...𝑁) × {0})) /
𝑝⦌𝐵 ≤ (𝑚 + 1))) |
418 | 320, 323,
416, 417 | syl3anbrc 1342 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...(𝑚 + 1))) |
419 | | fzss2 13295 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈
(ℤ≥‘(𝑚 + 1)) → (1...(𝑚 + 1)) ⊆ (1...𝑁)) |
420 | 291, 419 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...(𝑚 + 1)) ⊆ (1...𝑁)) |
421 | 420 | sselda 3926 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑛 ∈ (1...(𝑚 + 1))) → 𝑛 ∈ (1...𝑁)) |
422 | 421 | 3ad2antr1 1187 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → 𝑛 ∈ (1...𝑁)) |
423 | 354 | 3ad2antr2 1188 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
424 | 358 | ad2antll 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑞 Fn (1...(𝑚 + 1))) |
425 | 273 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) |
426 | | simprl 768 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑛 ∈ (1...(𝑚 + 1))) |
427 | | fvun1 6856 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
428 | 369, 427 | mp3an2 1448 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
429 | 424, 425,
426, 428 | syl12anc 834 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
430 | 429 | adantlrr 718 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
431 | 430 | 3adantr3 1170 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
432 | | simpr3 1195 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞‘𝑛) = 0) |
433 | 431, 432 | eqtrd 2780 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0) |
434 | 422, 423,
433 | 3jca 1127 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) |
435 | | nfv 1921 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) |
436 | | nfcv 2909 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑝𝑛 |
437 | 304, 377,
436 | nfbr 5126 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛 |
438 | 435, 437 | nfim 1903 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛) |
439 | | fveq1 6770 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘𝑛) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛)) |
440 | 439 | eqeq1d 2742 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) |
441 | 311, 440 | 3anbi23d 1438 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))) |
442 | 441 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)))) |
443 | 313 | breq1d 5089 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < 𝑛 ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)) |
444 | 442, 443 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛))) |
445 | 438, 310,
444, 221 | vtoclf 3496 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛) |
446 | 445 | adantlr 712 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛) |
447 | 434, 446 | syldan 591 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛) |
448 | | simp1 1135 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑛 ∈ (1...(𝑚 + 1))) |
449 | 421 | anasss 467 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → 𝑛 ∈ (1...𝑁)) |
450 | 448, 449 | sylanr2 680 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → 𝑛 ∈ (1...𝑁)) |
451 | | simp2 1136 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) |
452 | 451, 302 | sylanr2 680 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) |
453 | 429 | 3adantr3 1170 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛)) |
454 | | simpr3 1195 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → (𝑞‘𝑛) = 𝐾) |
455 | 453, 454 | eqtrd 2780 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾) |
456 | 455 | anasss 467 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾) |
457 | 456 | adantrlr 720 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾) |
458 | 450, 452,
457 | 3jca 1127 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) |
459 | | nfv 1921 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) |
460 | | nfcv 2909 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑝(𝑛 − 1) |
461 | 304, 460 | nfne 3047 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1) |
462 | 459, 461 | nfim 1903 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)) |
463 | 439 | eqeq1d 2742 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 𝐾 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) |
464 | 311, 463 | 3anbi23d 1438 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))) |
465 | 464 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)))) |
466 | 313 | neeq1d 3005 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ≠ (𝑛 − 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))) |
467 | 465, 466 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)))) |
468 | | poimirlem28.4 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) |
469 | 462, 310,
467, 468 | vtoclf 3496 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)) |
470 | 458, 469 | syldan 591 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)) |
471 | 470 | anassrs 468 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)) |
472 | 265, 267,
418, 447, 471 | poimirlem27 35800 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}))) |
473 | 265, 267,
418 | poimirlem26 35799 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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}))) |
474 | | fzfi 13690 |
. . . . . . . . . . . . . . . . . . 19
⊢
(0...(𝑚 + 1)) ∈
Fin |
475 | | xpfi 9063 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin ∧ (0...(𝑚 + 1)) ∈ Fin) →
((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin) |
476 | 254, 474,
475 | mp2an 689 |
. . . . . . . . . . . . . . . . . 18
⊢
((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin |
477 | | rabfi 9022 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin) |
478 | | hashcl 14069 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈
ℕ0) |
479 | 476, 477,
478 | mp2b 10 |
. . . . . . . . . . . . . . . . 17
⊢
(♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0 |
480 | 479 | nn0zi 12345 |
. . . . . . . . . . . . . . . 16
⊢
(♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ |
481 | | zsubcl 12362 |
. . . . . . . . . . . . . . . 16
⊢
(((♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ∈ ℤ) →
((♯‘{𝑡 ∈
((((0..^𝐾) ↑m
(1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))})) ∈ ℤ) |
482 | 480, 262,
481 | mp2an 689 |
. . . . . . . . . . . . . . 15
⊢
((♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))})) ∈ ℤ |
483 | | zsubcl 12362 |
. . . . . . . . . . . . . . . 16
⊢
(((♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ∈ ℤ) →
((♯‘{𝑡 ∈
((((0..^𝐾) ↑m
(1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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})) ∈ ℤ) |
484 | 480, 258,
483 | mp2an 689 |
. . . . . . . . . . . . . . 15
⊢
((♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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})) ∈ ℤ |
485 | | dvds2sub 15998 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℤ ∧ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))})) ∈ ℤ ∧
((♯‘{𝑡 ∈
((((0..^𝐾) ↑m
(1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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})) ∈ ℤ) → ((2 ∥
((♯‘{𝑡 ∈
((((0..^𝐾) ↑m
(1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))})) ∧ 2 ∥
((♯‘{𝑡 ∈
((((0..^𝐾) ↑m
(1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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}))) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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}))))) |
486 | 240, 482,
484, 485 | mp3an 1460 |
. . . . . . . . . . . . . 14
⊢ ((2
∥ ((♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))})) ∧ 2 ∥
((♯‘{𝑡 ∈
((((0..^𝐾) ↑m
(1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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}))) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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})))) |
487 | 472, 473,
486 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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})))) |
488 | 479 | nn0cni 12245 |
. . . . . . . . . . . . . 14
⊢
(♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ |
489 | 261 | nn0cni 12245 |
. . . . . . . . . . . . . 14
⊢
(♯‘{𝑠
∈ (((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ∈ ℂ |
490 | 257 | nn0cni 12245 |
. . . . . . . . . . . . . 14
⊢
(♯‘{𝑠
∈ (((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ∈ ℂ |
491 | | nnncan1 11257 |
. . . . . . . . . . . . . 14
⊢
(((♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ∈ ℂ ∧
(♯‘{𝑠 ∈
(((0..^𝐾) ↑m
(1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ∈ ℂ) →
(((♯‘{𝑡 ∈
((((0..^𝐾) ↑m
(1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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}))) = ((♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}))) |
492 | 488, 489,
490, 491 | mp3an 1460 |
. . . . . . . . . . . . 13
⊢
(((♯‘{𝑡
∈ ((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st
‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st
‘𝑠) ∘f
+ ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 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 + 1))∃𝑗 ∈ (0...(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}))) = ((♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))})) |
493 | 487, 492 | breqtrdi 5120 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}))) |
494 | | dvdssub2 16008 |
. . . . . . . . . . . 12
⊢ (((2
∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ∈ ℤ ∧
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ∈ ℤ) ∧ 2 ∥
((♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}))) → (2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}))) |
495 | 263, 493,
494 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}))) |
496 | | nn0cn 12243 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 ∈ ℕ0
→ 𝑚 ∈
ℂ) |
497 | | pncan1 11399 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚) |
498 | 496, 497 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈ ℕ0
→ ((𝑚 + 1) − 1)
= 𝑚) |
499 | 498 | oveq2d 7287 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ℕ0
→ (0...((𝑚 + 1)
− 1)) = (0...𝑚)) |
500 | 499 | rexeqdv 3348 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ℕ0
→ (∃𝑗 ∈
(0...((𝑚 + 1) −
1))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
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) |
501 | 499, 500 | raleqbidv 3335 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ0
→ (∀𝑖 ∈
(0...((𝑚 + 1) −
1))∃𝑗 ∈
(0...((𝑚 + 1) −
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 ∈ (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)) |
502 | 501 | 3anbi1d 1439 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ0
→ ((∀𝑖 ∈
(0...((𝑚 + 1) −
1))∃𝑗 ∈
(0...((𝑚 + 1) −
1))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
‘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 | 502 | rabbidv 3413 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ0
→ {𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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 + 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))}) |
504 | 503 | fveq2d 6775 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ0
→ (♯‘{𝑠
∈ (((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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 + 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))})) |
505 | 504 | ad2antrl 725 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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 + 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))})) |
506 | 1 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ ℕ) |
507 | 191 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝐾 ∈ ℕ) |
508 | | simprl 768 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 ∈ ℕ0) |
509 | | simprr 770 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 < 𝑁) |
510 | 506, 507,
508, 509 | poimirlem4 35777 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 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} ≈ {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))}) |
511 | | fzfi 13690 |
. . . . . . . . . . . . . . . . . 18
⊢
(1...𝑚) ∈
Fin |
512 | | mapfi 9093 |
. . . . . . . . . . . . . . . . . 18
⊢
(((0..^𝐾) ∈ Fin
∧ (1...𝑚) ∈ Fin)
→ ((0..^𝐾)
↑m (1...𝑚))
∈ Fin) |
513 | 10, 511, 512 | mp2an 689 |
. . . . . . . . . . . . . . . . 17
⊢
((0..^𝐾)
↑m (1...𝑚))
∈ Fin |
514 | | ovex 7304 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(1...𝑚) ∈
V |
515 | 514, 514 | mapval 8610 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1...𝑚)
↑m (1...𝑚))
= {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} |
516 | | mapfi 9093 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1...𝑚) ∈ Fin
∧ (1...𝑚) ∈ Fin)
→ ((1...𝑚)
↑m (1...𝑚))
∈ Fin) |
517 | 511, 511,
516 | mp2an 689 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1...𝑚)
↑m (1...𝑚))
∈ Fin |
518 | 515, 517 | eqeltrri 2838 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin |
519 | | f1of 6714 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:(1...𝑚)–1-1-onto→(1...𝑚) → 𝑓:(1...𝑚)⟶(1...𝑚)) |
520 | 519 | ss2abi 4005 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} |
521 | | ssfi 8938 |
. . . . . . . . . . . . . . . . . 18
⊢ (({𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}) → {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin) |
522 | 518, 520,
521 | mp2an 689 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin |
523 | | xpfi 9063 |
. . . . . . . . . . . . . . . . 17
⊢
((((0..^𝐾)
↑m (1...𝑚))
∈ Fin ∧ {𝑓 ∣
𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin) → (((0..^𝐾) ↑m (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin) |
524 | 513, 522,
523 | mp2an 689 |
. . . . . . . . . . . . . . . 16
⊢
(((0..^𝐾)
↑m (1...𝑚))
× {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin |
525 | | rabfi 9022 |
. . . . . . . . . . . . . . . 16
⊢
((((0..^𝐾)
↑m (1...𝑚))
× {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin → {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} ∈ Fin) |
526 | 524, 525 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ {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} ∈ Fin |
527 | | rabfi 9022 |
. . . . . . . . . . . . . . . 16
⊢
((((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {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))} ∈ Fin) |
528 | 254, 527 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ {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))} ∈ Fin |
529 | | hashen 14059 |
. . . . . . . . . . . . . . 15
⊢ (({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} ∈ Fin ∧ {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))} ∈ Fin) →
((♯‘{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))}) ↔ {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))})) |
530 | 526, 528,
529 | mp2an 689 |
. . . . . . . . . . . . . 14
⊢
((♯‘{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))}) ↔ {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))}) |
531 | 510, 530 | sylibr 233 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 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}) = (♯‘{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))})) |
532 | 505, 531 | eqtr4d 2783 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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})) |
533 | 532 | breq2d 5091 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))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))}) ↔ 2 ∥
(♯‘{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}))) |
534 | 495, 533 | bitrd 278 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) ↔ 2 ∥ (♯‘{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}))) |
535 | 534 | biimpd 228 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}) → 2 ∥ (♯‘{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}))) |
536 | 535 | con3d 152 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (¬ 2 ∥
(♯‘{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}) → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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}))) |
537 | 536 | expcom 414 |
. . . . . . 7
⊢ ((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) → (𝜑 → (¬ 2 ∥
(♯‘{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}) → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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})))) |
538 | 537 | a2d 29 |
. . . . . 6
⊢ ((𝑚 ∈ ℕ0
∧ 𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥
(♯‘{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})) → (𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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})))) |
539 | 538 | 3adant1 1129 |
. . . . 5
⊢ ((𝑁 ∈ ℕ0
∧ 𝑚 ∈
ℕ0 ∧ 𝑚
< 𝑁) → ((𝜑 → ¬ 2 ∥
(♯‘{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})) → (𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...(𝑚
+ 1))) × {𝑓 ∣
𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(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})))) |
540 | 107, 132,
157, 182, 239, 539 | fnn0ind 12419 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑁
≤ 𝑁) → (𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
541 | 5, 540 | mpcom 38 |
. . 3
⊢ (𝜑 → ¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) |
542 | | dvds0 15979 |
. . . . . . . 8
⊢ (2 ∈
ℤ → 2 ∥ 0) |
543 | 240, 542 | ax-mp 5 |
. . . . . . 7
⊢ 2 ∥
0 |
544 | | hash0 14080 |
. . . . . . 7
⊢
(♯‘∅) = 0 |
545 | 543, 544 | breqtrri 5106 |
. . . . . 6
⊢ 2 ∥
(♯‘∅) |
546 | | fveq2 6771 |
. . . . . 6
⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) =
(♯‘∅)) |
547 | 545, 546 | breqtrrid 5117 |
. . . . 5
⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})) |
548 | 3 | ltp1d 11905 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 < (𝑁 + 1)) |
549 | 282 | peano2zd 12428 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
550 | | fzn 13271 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅)) |
551 | 549, 282,
550 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅)) |
552 | 548, 551 | mpbid 231 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅) |
553 | 552 | xpeq1d 5619 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = (∅ ×
{0})) |
554 | 553, 86 | eqtrdi 2796 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = ∅) |
555 | 554 | uneq2d 4102 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪
∅)) |
556 | | un0 4330 |
. . . . . . . . . . . . . 14
⊢
(((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅)
= ((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) ×
{0}))) |
557 | 555, 556 | eqtrdi 2796 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = ((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))) |
558 | 557 | csbeq1d 3841 |
. . . . . . . . . . . 12
⊢ (𝜑 →
⦋(((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵) |
559 | | ovex 7304 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0}))) ∈
V |
560 | | poimirlem28.1 |
. . . . . . . . . . . . 13
⊢ (𝑝 = ((1st ‘𝑠) ∘f +
((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) |
561 | 559, 560 | csbie 3873 |
. . . . . . . . . . . 12
⊢
⦋((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = 𝐶 |
562 | 558, 561 | eqtrdi 2796 |
. . . . . . . . . . 11
⊢ (𝜑 →
⦋(((1st ‘𝑠) ∘f + ((((2nd
‘𝑠) “
(1...𝑗)) × {1}) ∪
(((2nd ‘𝑠)
“ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = 𝐶) |
563 | 562 | eqeq2d 2751 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = 𝐶)) |
564 | 563 | rexbidv 3228 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
565 | 564 | ralbidv 3123 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) |
566 | 565 | rabbidv 3413 |
. . . . . . 7
⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) |
567 | 566 | fveq2d 6775 |
. . . . . 6
⊢ (𝜑 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})) |
568 | 567 | breq2d 5091 |
. . . . 5
⊢ (𝜑 → (2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))) |
569 | 547, 568 | syl5ibr 245 |
. . . 4
⊢ (𝜑 → ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) |
570 | 569 | necon3bd 2959 |
. . 3
⊢ (𝜑 → (¬ 2 ∥
(♯‘{𝑠 ∈
(((0..^𝐾)
↑m (1...𝑁))
× {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st
‘𝑠)
∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅)) |
571 | 541, 570 | mpd 15 |
. 2
⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅) |
572 | | rabn0 4325 |
. 2
⊢ ({𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅ ↔ ∃𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) |
573 | 571, 572 | sylib 217 |
1
⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) |