Step | Hyp | Ref
| Expression |
1 | | simpll 524 |
. . . . . . . . 9
⊢ (((𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → 𝐴 ⊆ (ℤ≥‘𝑚)) |
2 | | simprr 527 |
. . . . . . . . 9
⊢ (((𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) |
3 | 1, 2 | jca 304 |
. . . . . . . 8
⊢ (((𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
4 | | nfcv 2312 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖if(𝑘 ∈ 𝐴, 𝐵, 1) |
5 | | nfv 1521 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘 𝑖 ∈ 𝐴 |
6 | | nfcsb1v 3082 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 |
7 | | nfcv 2312 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘1 |
8 | 5, 6, 7 | nfif 3553 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1) |
9 | | eleq1w 2231 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)) |
10 | | csbeq1a 3058 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵) |
11 | 9, 10 | ifbieq1d 3547 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑖 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1)) |
12 | 4, 8, 11 | cbvmpt 4082 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑖 ∈ ℤ ↦ if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1)) |
13 | | simpll 524 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝜑) |
14 | | zprod.6 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
15 | 14 | ralrimiva 2543 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
16 | 6 | nfel1 2323 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ |
17 | 10 | eleq1d 2239 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)) |
18 | 16, 17 | rspc 2828 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)) |
19 | 15, 18 | syl5 32 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ 𝐴 → (𝜑 → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)) |
20 | 13, 19 | mpan9 279 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ) |
21 | | simplr 525 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝑚 ∈ ℤ) |
22 | | zprod.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) |
23 | 22 | ad2antrr 485 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝑀 ∈ ℤ) |
24 | | simpr 109 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝐴 ⊆ (ℤ≥‘𝑚)) |
25 | | zprod.4 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ⊆ 𝑍) |
26 | | zprod.1 |
. . . . . . . . . . . . 13
⊢ 𝑍 =
(ℤ≥‘𝑀) |
27 | 25, 26 | sseqtrdi 3195 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) |
28 | 27 | ad2antrr 485 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝐴 ⊆ (ℤ≥‘𝑀)) |
29 | | zproddc.dc |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴) |
30 | 26 | raleqi 2669 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑗 ∈
𝑍 DECID
𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) |
31 | 29, 30 | sylib 121 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) |
32 | | eleq1w 2231 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴)) |
33 | 32 | dcbid 833 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑖 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑖 ∈ 𝐴)) |
34 | 33 | cbvralv 2696 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑗 ∈
(ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑖 ∈ (ℤ≥‘𝑀)DECID 𝑖 ∈ 𝐴) |
35 | 31, 34 | sylib 121 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑖 ∈ (ℤ≥‘𝑀)DECID 𝑖 ∈ 𝐴) |
36 | 35 | r19.21bi 2558 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → DECID
𝑖 ∈ 𝐴) |
37 | 36 | adantlr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → DECID
𝑖 ∈ 𝐴) |
38 | 37 | adantlr 474 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → DECID
𝑖 ∈ 𝐴) |
39 | 38 | adantlr 474 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆
(ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → DECID
𝑖 ∈ 𝐴) |
40 | | simp-4l 536 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆
(ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) ∧ ¬ 𝑖 ∈
(ℤ≥‘𝑀)) → 𝜑) |
41 | | simpr 109 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆
(ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) ∧ ¬ 𝑖 ∈
(ℤ≥‘𝑀)) → ¬ 𝑖 ∈ (ℤ≥‘𝑀)) |
42 | 27 | ssneld 3149 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (¬ 𝑖 ∈ (ℤ≥‘𝑀) → ¬ 𝑖 ∈ 𝐴)) |
43 | 40, 41, 42 | sylc 62 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆
(ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) ∧ ¬ 𝑖 ∈
(ℤ≥‘𝑀)) → ¬ 𝑖 ∈ 𝐴) |
44 | 43 | olcd 729 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆
(ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) ∧ ¬ 𝑖 ∈
(ℤ≥‘𝑀)) → (𝑖 ∈ 𝐴 ∨ ¬ 𝑖 ∈ 𝐴)) |
45 | | df-dc 830 |
. . . . . . . . . . . . 13
⊢
(DECID 𝑖 ∈ 𝐴 ↔ (𝑖 ∈ 𝐴 ∨ ¬ 𝑖 ∈ 𝐴)) |
46 | 44, 45 | sylibr 133 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆
(ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) ∧ ¬ 𝑖 ∈
(ℤ≥‘𝑀)) → DECID 𝑖 ∈ 𝐴) |
47 | | eluzelz 9483 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈
(ℤ≥‘𝑚) → 𝑖 ∈ ℤ) |
48 | | eluzdc 9556 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) →
DECID 𝑖
∈ (ℤ≥‘𝑀)) |
49 | 23, 47, 48 | syl2an 287 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) → DECID
𝑖 ∈
(ℤ≥‘𝑀)) |
50 | | exmiddc 831 |
. . . . . . . . . . . . 13
⊢
(DECID 𝑖 ∈ (ℤ≥‘𝑀) → (𝑖 ∈ (ℤ≥‘𝑀) ∨ ¬ 𝑖 ∈ (ℤ≥‘𝑀))) |
51 | 49, 50 | syl 14 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) → (𝑖 ∈ (ℤ≥‘𝑀) ∨ ¬ 𝑖 ∈ (ℤ≥‘𝑀))) |
52 | 39, 46, 51 | mpjaodan 793 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) ∧ 𝑖 ∈ (ℤ≥‘𝑚)) → DECID
𝑖 ∈ 𝐴) |
53 | 12, 20, 21, 23, 24, 28, 52, 38 | prodrbdc 11524 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
54 | 53 | biimpd 143 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
55 | 54 | expimpd 361 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
56 | 3, 55 | syl5 32 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
57 | 56 | rexlimdva 2587 |
. . . . . 6
⊢ (𝜑 → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
58 | | uzssz 9493 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
59 | 27, 58 | sstrdi 3159 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ⊆ ℤ) |
60 | 59 | ad2antrr 485 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ⊆ ℤ) |
61 | | 1zzd 9226 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 1 ∈
ℤ) |
62 | | nnz 9218 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℤ) |
63 | 62 | adantl 275 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ) |
64 | 63 | adantr 274 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ ℤ) |
65 | 61, 64 | fzfigd 10374 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ∈ Fin) |
66 | | simpr 109 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑓:(1...𝑚)–1-1-onto→𝐴) |
67 | | f1oeng 6731 |
. . . . . . . . . . . . . . 15
⊢
(((1...𝑚) ∈ Fin
∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴) |
68 | 65, 66, 67 | syl2anc 409 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴) |
69 | 68 | ensymd 6757 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ≈ (1...𝑚)) |
70 | | enfii 6848 |
. . . . . . . . . . . . 13
⊢
(((1...𝑚) ∈ Fin
∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin) |
71 | 65, 69, 70 | syl2anc 409 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ∈ Fin) |
72 | | zfz1iso 10763 |
. . . . . . . . . . . 12
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴)) |
73 | 60, 71, 72 | syl2anc 409 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴)) |
74 | | simpll 524 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑) |
75 | 74, 19 | mpan9 279 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ) |
76 | | breq1 3990 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴))) |
77 | | fveq2 5494 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗)) |
78 | 77 | csbeq1d 3056 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵) |
79 | 76, 78 | ifbieq1d 3547 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1)) |
80 | | csbcow 3060 |
. . . . . . . . . . . . . . . . . 18
⊢
⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 |
81 | | ifeq1 3528 |
. . . . . . . . . . . . . . . . . 18
⊢
(⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 → if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1)) |
82 | 80, 81 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1) |
83 | 79, 82 | eqtr4di 2221 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1)) |
84 | 83 | cbvmptv 4083 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1)) |
85 | | eqid 2170 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1)) |
86 | 36 | ad4ant14 511 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → DECID
𝑖 ∈ 𝐴) |
87 | | simplr 525 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ) |
88 | 22 | ad2antrr 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ) |
89 | 27 | ad2antrr 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑀)) |
90 | | simprl 526 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴) |
91 | | simprr 527 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴)) |
92 | 12, 75, 84, 85, 86, 87, 88, 89, 90, 91 | prodmodclem2a 11526 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) |
93 | 65 | adantrr 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (1...𝑚) ∈ Fin) |
94 | 93, 90 | fihasheqf1od 10711 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘(1...𝑚)) = (♯‘𝐴)) |
95 | 87 | nnnn0d 9175 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ0) |
96 | | hashfz1 10704 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ ℕ0
→ (♯‘(1...𝑚)) = 𝑚) |
97 | 95, 96 | syl 14 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘(1...𝑚)) = 𝑚) |
98 | 94, 97 | eqtr3d 2205 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘𝐴) = 𝑚) |
99 | 98 | breq2d 3999 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛 ≤ 𝑚)) |
100 | 99 | ifbid 3546 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1) = if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)) |
101 | 100 | mpteq2dv 4078 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))) |
102 | 101 | seqeq3d 10396 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))) |
103 | 102 | fveq1d 5496 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) |
104 | 92, 103 | breqtrd 4013 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) |
105 | 104 | expr 373 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) |
106 | 105 | exlimdv 1812 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) |
107 | 73, 106 | mpd 13 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) |
108 | | breq2 3991 |
. . . . . . . . . 10
⊢ (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚) → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) |
109 | 107, 108 | syl5ibrcom 156 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
110 | 109 | expimpd 361 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
111 | 110 | exlimdv 1812 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
112 | 111 | rexlimdva 2587 |
. . . . . 6
⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
113 | 57, 112 | jaod 712 |
. . . . 5
⊢ (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
114 | 22 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝑀 ∈ ℤ) |
115 | 27 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝐴 ⊆ (ℤ≥‘𝑀)) |
116 | 31 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) |
117 | 115, 116 | jca 304 |
. . . . . . . 8
⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → (𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈
(ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)) |
118 | | zproddc.3 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦)) |
119 | 26 | eleq2i 2237 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀)) |
120 | | eluzelz 9483 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → 𝑛 ∈ ℤ) |
121 | 120 | adantl 275 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ ℤ) |
122 | | simpr 109 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → 𝑝 ∈ (ℤ≥‘𝑛)) |
123 | | simplr 525 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
124 | | uztrn 9490 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑝 ∈
(ℤ≥‘𝑛) ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑝 ∈ (ℤ≥‘𝑀)) |
125 | 122, 123,
124 | syl2anc 409 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → 𝑝 ∈ (ℤ≥‘𝑀)) |
126 | 125, 26 | eleqtrrdi 2264 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → 𝑝 ∈ 𝑍) |
127 | | zprod.5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
128 | 127 | ralrimiva 2543 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
129 | 128 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
130 | | nfv 1521 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘 𝑝 ∈ 𝐴 |
131 | | nfcsb1v 3082 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘⦋𝑝 / 𝑘⦌𝐵 |
132 | 130, 131,
7 | nfif 3553 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) |
133 | 132 | nfeq2 2324 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(𝐹‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) |
134 | | fveq2 5494 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑝 → (𝐹‘𝑘) = (𝐹‘𝑝)) |
135 | | eleq1w 2231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑝 → (𝑘 ∈ 𝐴 ↔ 𝑝 ∈ 𝐴)) |
136 | | csbeq1a 3058 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑝 → 𝐵 = ⦋𝑝 / 𝑘⦌𝐵) |
137 | 135, 136 | ifbieq1d 3547 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑝 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)) |
138 | 134, 137 | eqeq12d 2185 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑝 → ((𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1) ↔ (𝐹‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))) |
139 | 133, 138 | rspc 2828 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1) → (𝐹‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))) |
140 | 126, 129,
139 | sylc 62 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)) |
141 | | simpr 109 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐴) |
142 | 15 | ad3antrrr 489 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) ∧ 𝑝 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
143 | 131 | nfel1 2323 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ |
144 | 136 | eleq1d 2239 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑝 → (𝐵 ∈ ℂ ↔ ⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ)) |
145 | 143, 144 | rspc 2828 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ)) |
146 | 141, 142,
145 | sylc 62 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) ∧ 𝑝 ∈ 𝐴) → ⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ) |
147 | | 1cnd 7923 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) ∧ ¬ 𝑝 ∈ 𝐴) → 1 ∈ ℂ) |
148 | | eleq1w 2231 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑝 → (𝑗 ∈ 𝐴 ↔ 𝑝 ∈ 𝐴)) |
149 | 148 | dcbid 833 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑝 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑝 ∈ 𝐴)) |
150 | 29 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴) |
151 | 149, 150,
126 | rspcdva 2839 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → DECID
𝑝 ∈ 𝐴) |
152 | 146, 147,
151 | ifcldadc 3554 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ) |
153 | 140, 152 | eqeltrd 2247 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑝) ∈ ℂ) |
154 | | simpr 109 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ (ℤ≥‘𝑛)) |
155 | | simplr 525 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
156 | | uztrn 9490 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑟 ∈
(ℤ≥‘𝑛) ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑟 ∈ (ℤ≥‘𝑀)) |
157 | 154, 155,
156 | syl2anc 409 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ (ℤ≥‘𝑀)) |
158 | 157, 26 | eleqtrrdi 2264 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ 𝑍) |
159 | 128 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
160 | | nfv 1521 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘 𝑟 ∈ 𝐴 |
161 | | nfcsb1v 3082 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘⦋𝑟 / 𝑘⦌𝐵 |
162 | 160, 161,
7 | nfif 3553 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑘if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1) |
163 | 162 | nfeq2 2324 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘(𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1) |
164 | | fveq2 5494 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑟 → (𝐹‘𝑘) = (𝐹‘𝑟)) |
165 | | eleq1w 2231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑟 → (𝑘 ∈ 𝐴 ↔ 𝑟 ∈ 𝐴)) |
166 | | csbeq1a 3058 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑟 → 𝐵 = ⦋𝑟 / 𝑘⦌𝐵) |
167 | 165, 166 | ifbieq1d 3547 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑟 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)) |
168 | 164, 167 | eqeq12d 2185 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑟 → ((𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1) ↔ (𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1))) |
169 | 163, 168 | rspc 2828 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1) → (𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1))) |
170 | 158, 159,
169 | sylc 62 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)) |
171 | 58, 157 | sselid 3145 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ ℤ) |
172 | | simpr 109 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴) |
173 | 15 | ad3antrrr 489 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) ∧ 𝑟 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
174 | 161 | nfel1 2323 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑘⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ |
175 | 166 | eleq1d 2239 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑟 → (𝐵 ∈ ℂ ↔ ⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ)) |
176 | 174, 175 | rspc 2828 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ)) |
177 | 172, 173,
176 | sylc 62 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) ∧ 𝑟 ∈ 𝐴) → ⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ) |
178 | | 1cnd 7923 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) ∧ ¬ 𝑟 ∈ 𝐴) → 1 ∈ ℂ) |
179 | | eleq1w 2231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑟 → (𝑗 ∈ 𝐴 ↔ 𝑟 ∈ 𝐴)) |
180 | 179 | dcbid 833 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑟 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑟 ∈ 𝐴)) |
181 | 29 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴) |
182 | 180, 181,
158 | rspcdva 2839 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → DECID
𝑟 ∈ 𝐴) |
183 | 177, 178,
182 | ifcldadc 3554 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1) ∈ ℂ) |
184 | | nfcv 2312 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑘𝑟 |
185 | | eqid 2170 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) |
186 | 184, 162,
167, 185 | fvmptf 5586 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑟 ∈ ℤ ∧ if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1) ∈ ℂ) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)) |
187 | 171, 183,
186 | syl2anc 409 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)) |
188 | 170, 187 | eqtr4d 2206 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑟) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟)) |
189 | | mulcl 7888 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝 · 𝑞) ∈ ℂ) |
190 | 189 | adantl 275 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) ∧ (𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ)) → (𝑝 · 𝑞) ∈ ℂ) |
191 | 121, 153,
188, 190 | seq3feq 10415 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → seq𝑛( · , 𝐹) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))) |
192 | 191 | breq1d 3997 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)) |
193 | 192 | anbi2d 461 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))) |
194 | 193 | exbidv 1818 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))) |
195 | 119, 194 | sylan2b 285 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))) |
196 | 195 | rexbidva 2467 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))) |
197 | 118, 196 | mpbid 146 |
. . . . . . . . . 10
⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)) |
198 | 26 | rexeqi 2670 |
. . . . . . . . . 10
⊢
(∃𝑛 ∈
𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)) |
199 | 197, 198 | sylib 121 |
. . . . . . . . 9
⊢ (𝜑 → ∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)) |
200 | 199 | anim1i 338 |
. . . . . . . 8
⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → (∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
201 | | fveq2 5494 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) =
(ℤ≥‘𝑀)) |
202 | 201 | sseq2d 3177 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑀))) |
203 | 201 | raleqdv 2671 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑀 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)) |
204 | 202, 203 | anbi12d 470 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑀 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ↔ (𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈
(ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴))) |
205 | 201 | rexeqdv 2672 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))) |
206 | | seqeq1 10391 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑀 → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))) |
207 | 206 | breq1d 3997 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑀 → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
208 | 205, 207 | anbi12d 470 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑀 → ((∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))) |
209 | 204, 208 | anbi12d 470 |
. . . . . . . . 9
⊢ (𝑚 = 𝑀 → (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈
(ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)))) |
210 | 209 | rspcev 2834 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ ((𝐴 ⊆
(ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))) → ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))) |
211 | 114, 117,
200, 210 | syl12anc 1231 |
. . . . . . 7
⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))) |
212 | 211 | orcd 728 |
. . . . . 6
⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))) |
213 | 212 | ex 114 |
. . . . 5
⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))) |
214 | 113, 213 | impbid 128 |
. . . 4
⊢ (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) |
215 | | eluzelz 9483 |
. . . . . . . 8
⊢ (𝑝 ∈
(ℤ≥‘𝑀) → 𝑝 ∈ ℤ) |
216 | | simpr 109 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐴) |
217 | 15 | ad2antrr 485 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
218 | 216, 217,
145 | sylc 62 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) ∧ 𝑝 ∈ 𝐴) → ⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ) |
219 | | 1cnd 7923 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑝 ∈ 𝐴) → 1 ∈ ℂ) |
220 | 29 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴) |
221 | 26 | eleq2i 2237 |
. . . . . . . . . . . 12
⊢ (𝑝 ∈ 𝑍 ↔ 𝑝 ∈ (ℤ≥‘𝑀)) |
222 | 221 | biimpri 132 |
. . . . . . . . . . 11
⊢ (𝑝 ∈
(ℤ≥‘𝑀) → 𝑝 ∈ 𝑍) |
223 | 222 | adantl 275 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) → 𝑝 ∈ 𝑍) |
224 | 149, 220,
223 | rspcdva 2839 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) → DECID
𝑝 ∈ 𝐴) |
225 | 218, 219,
224 | ifcldadc 3554 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ) |
226 | | nfcv 2312 |
. . . . . . . . 9
⊢
Ⅎ𝑘𝑝 |
227 | 226, 132,
137, 185 | fvmptf 5586 |
. . . . . . . 8
⊢ ((𝑝 ∈ ℤ ∧ if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)) |
228 | 215, 225,
227 | syl2an2 589 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)) |
229 | 228, 225 | eqeltrd 2247 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) ∈ ℂ) |
230 | | eluzelz 9483 |
. . . . . . . 8
⊢ (𝑟 ∈
(ℤ≥‘𝑀) → 𝑟 ∈ ℤ) |
231 | | simpr 109 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴) |
232 | 15 | ad2antrr 485 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
233 | 231, 232,
176 | sylc 62 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) ∧ 𝑟 ∈ 𝐴) → ⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ) |
234 | | 1cnd 7923 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑟 ∈ 𝐴) → 1 ∈ ℂ) |
235 | 29 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴) |
236 | 26 | eleq2i 2237 |
. . . . . . . . . . . 12
⊢ (𝑟 ∈ 𝑍 ↔ 𝑟 ∈ (ℤ≥‘𝑀)) |
237 | 236 | biimpri 132 |
. . . . . . . . . . 11
⊢ (𝑟 ∈
(ℤ≥‘𝑀) → 𝑟 ∈ 𝑍) |
238 | 237 | adantl 275 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) → 𝑟 ∈ 𝑍) |
239 | 180, 235,
238 | rspcdva 2839 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) → DECID
𝑟 ∈ 𝐴) |
240 | 233, 234,
239 | ifcldadc 3554 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) → if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1) ∈ ℂ) |
241 | 230, 240,
186 | syl2an2 589 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)) |
242 | 128 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1)) |
243 | 238, 242,
169 | sylc 62 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)) |
244 | 241, 243 | eqtr4d 2206 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟) = (𝐹‘𝑟)) |
245 | 189 | adantl 275 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ)) → (𝑝 · 𝑞) ∈ ℂ) |
246 | 22, 229, 244, 245 | seq3feq 10415 |
. . . . 5
⊢ (𝜑 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑀( · , 𝐹)) |
247 | 246 | breq1d 3997 |
. . . 4
⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , 𝐹) ⇝ 𝑥)) |
248 | 214, 247 | bitrd 187 |
. . 3
⊢ (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) ↔ seq𝑀( · , 𝐹) ⇝ 𝑥)) |
249 | 248 | iotabidv 5179 |
. 2
⊢ (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥)) |
250 | | df-proddc 11501 |
. 2
⊢
∏𝑘 ∈
𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈
(ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))) |
251 | | df-fv 5204 |
. 2
⊢ ( ⇝
‘seq𝑀( · ,
𝐹)) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥) |
252 | 249, 250,
251 | 3eqtr4g 2228 |
1
⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹))) |