Step | Hyp | Ref
| Expression |
1 | | prodmo.1 |
. . 3
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) |
2 | | prodmo.2 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
3 | | prodmodclem2a.dc |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID
𝑘 ∈ 𝐴) |
4 | | prodmolem2.7 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀)) |
5 | | prodmolem2.9 |
. . . . . . 7
⊢ (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴)) |
6 | | 1zzd 9194 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℤ) |
7 | | prodmolem2.5 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℕ) |
8 | 7 | nnzd 9285 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℤ) |
9 | 6, 8 | fzfigd 10330 |
. . . . . . . . . . 11
⊢ (𝜑 → (1...𝑁) ∈ Fin) |
10 | | prodmolem2.8 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴) |
11 | 9, 10 | fihasheqf1od 10664 |
. . . . . . . . . 10
⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘𝐴)) |
12 | 7 | nnnn0d 9143 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
13 | | hashfz1 10657 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ0
→ (♯‘(1...𝑁)) = 𝑁) |
14 | 12, 13 | syl 14 |
. . . . . . . . . 10
⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁) |
15 | 11, 14 | eqtr3d 2192 |
. . . . . . . . 9
⊢ (𝜑 → (♯‘𝐴) = 𝑁) |
16 | 15 | oveq2d 5840 |
. . . . . . . 8
⊢ (𝜑 → (1...(♯‘𝐴)) = (1...𝑁)) |
17 | | isoeq4 5754 |
. . . . . . . 8
⊢
((1...(♯‘𝐴)) = (1...𝑁) → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴))) |
18 | 16, 17 | syl 14 |
. . . . . . 7
⊢ (𝜑 → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴))) |
19 | 5, 18 | mpbid 146 |
. . . . . 6
⊢ (𝜑 → 𝐾 Isom < , < ((1...𝑁), 𝐴)) |
20 | | isof1o 5757 |
. . . . . 6
⊢ (𝐾 Isom < , < ((1...𝑁), 𝐴) → 𝐾:(1...𝑁)–1-1-onto→𝐴) |
21 | | f1of 5414 |
. . . . . 6
⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴) |
22 | 19, 20, 21 | 3syl 17 |
. . . . 5
⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴) |
23 | | nnuz 9474 |
. . . . . . 7
⊢ ℕ =
(ℤ≥‘1) |
24 | 7, 23 | eleqtrdi 2250 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘1)) |
25 | | eluzfz2 9934 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘1) → 𝑁 ∈ (1...𝑁)) |
26 | 24, 25 | syl 14 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ (1...𝑁)) |
27 | 22, 26 | ffvelrnd 5603 |
. . . 4
⊢ (𝜑 → (𝐾‘𝑁) ∈ 𝐴) |
28 | 4, 27 | sseldd 3129 |
. . 3
⊢ (𝜑 → (𝐾‘𝑁) ∈ (ℤ≥‘𝑀)) |
29 | 4 | sselda 3128 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (ℤ≥‘𝑀)) |
30 | 19, 20 | syl 14 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴) |
31 | | f1ocnvfv2 5728 |
. . . . . . . . 9
⊢ ((𝐾:(1...𝑁)–1-1-onto→𝐴 ∧ 𝑝 ∈ 𝐴) → (𝐾‘(◡𝐾‘𝑝)) = 𝑝) |
32 | 30, 31 | sylan 281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝐾‘(◡𝐾‘𝑝)) = 𝑝) |
33 | | f1ocnv 5427 |
. . . . . . . . . . . 12
⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → ◡𝐾:𝐴–1-1-onto→(1...𝑁)) |
34 | | f1of 5414 |
. . . . . . . . . . . 12
⊢ (◡𝐾:𝐴–1-1-onto→(1...𝑁) → ◡𝐾:𝐴⟶(1...𝑁)) |
35 | 30, 33, 34 | 3syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ◡𝐾:𝐴⟶(1...𝑁)) |
36 | 35 | ffvelrnda 5602 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (◡𝐾‘𝑝) ∈ (1...𝑁)) |
37 | | elfzle2 9930 |
. . . . . . . . . 10
⊢ ((◡𝐾‘𝑝) ∈ (1...𝑁) → (◡𝐾‘𝑝) ≤ 𝑁) |
38 | 36, 37 | syl 14 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (◡𝐾‘𝑝) ≤ 𝑁) |
39 | 19 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐾 Isom < , < ((1...𝑁), 𝐴)) |
40 | | fzssuz 9967 |
. . . . . . . . . . . . 13
⊢
(1...𝑁) ⊆
(ℤ≥‘1) |
41 | | uzssz 9458 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘1) ⊆ ℤ |
42 | | zssre 9174 |
. . . . . . . . . . . . . 14
⊢ ℤ
⊆ ℝ |
43 | 41, 42 | sstri 3137 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘1) ⊆ ℝ |
44 | 40, 43 | sstri 3137 |
. . . . . . . . . . . 12
⊢
(1...𝑁) ⊆
ℝ |
45 | | ressxr 7921 |
. . . . . . . . . . . 12
⊢ ℝ
⊆ ℝ* |
46 | 44, 45 | sstri 3137 |
. . . . . . . . . . 11
⊢
(1...𝑁) ⊆
ℝ* |
47 | 46 | a1i 9 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (1...𝑁) ⊆
ℝ*) |
48 | | uzssz 9458 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
49 | 48, 42 | sstri 3137 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘𝑀) ⊆ ℝ |
50 | 49, 45 | sstri 3137 |
. . . . . . . . . . . 12
⊢
(ℤ≥‘𝑀) ⊆
ℝ* |
51 | 4, 50 | sstrdi 3140 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ⊆
ℝ*) |
52 | 51 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐴 ⊆
ℝ*) |
53 | 26 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑁 ∈ (1...𝑁)) |
54 | | leisorel 10708 |
. . . . . . . . . 10
⊢ ((𝐾 Isom < , < ((1...𝑁), 𝐴) ∧ ((1...𝑁) ⊆ ℝ* ∧ 𝐴 ⊆ ℝ*)
∧ ((◡𝐾‘𝑝) ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → ((◡𝐾‘𝑝) ≤ 𝑁 ↔ (𝐾‘(◡𝐾‘𝑝)) ≤ (𝐾‘𝑁))) |
55 | 39, 47, 52, 36, 53, 54 | syl122anc 1229 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → ((◡𝐾‘𝑝) ≤ 𝑁 ↔ (𝐾‘(◡𝐾‘𝑝)) ≤ (𝐾‘𝑁))) |
56 | 38, 55 | mpbid 146 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝐾‘(◡𝐾‘𝑝)) ≤ (𝐾‘𝑁)) |
57 | 32, 56 | eqbrtrrd 3988 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ≤ (𝐾‘𝑁)) |
58 | 4, 48 | sstrdi 3140 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℤ) |
59 | 58 | sselda 3128 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ ℤ) |
60 | 48, 28 | sseldi 3126 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾‘𝑁) ∈ ℤ) |
61 | 60 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝐾‘𝑁) ∈ ℤ) |
62 | | eluz 9452 |
. . . . . . . 8
⊢ ((𝑝 ∈ ℤ ∧ (𝐾‘𝑁) ∈ ℤ) → ((𝐾‘𝑁) ∈ (ℤ≥‘𝑝) ↔ 𝑝 ≤ (𝐾‘𝑁))) |
63 | 59, 61, 62 | syl2anc 409 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → ((𝐾‘𝑁) ∈ (ℤ≥‘𝑝) ↔ 𝑝 ≤ (𝐾‘𝑁))) |
64 | 57, 63 | mpbird 166 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝐾‘𝑁) ∈ (ℤ≥‘𝑝)) |
65 | | elfzuzb 9922 |
. . . . . 6
⊢ (𝑝 ∈ (𝑀...(𝐾‘𝑁)) ↔ (𝑝 ∈ (ℤ≥‘𝑀) ∧ (𝐾‘𝑁) ∈ (ℤ≥‘𝑝))) |
66 | 29, 64, 65 | sylanbrc 414 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (𝑀...(𝐾‘𝑁))) |
67 | 66 | ex 114 |
. . . 4
⊢ (𝜑 → (𝑝 ∈ 𝐴 → 𝑝 ∈ (𝑀...(𝐾‘𝑁)))) |
68 | 67 | ssrdv 3134 |
. . 3
⊢ (𝜑 → 𝐴 ⊆ (𝑀...(𝐾‘𝑁))) |
69 | 1, 2, 3, 28, 68 | fproddccvg 11469 |
. 2
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq𝑀( · , 𝐹)‘(𝐾‘𝑁))) |
70 | | mulid2 7876 |
. . . . 5
⊢ (𝑚 ∈ ℂ → (1
· 𝑚) = 𝑚) |
71 | 70 | adantl 275 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (1 · 𝑚) = 𝑚) |
72 | | mulid1 7875 |
. . . . 5
⊢ (𝑚 ∈ ℂ → (𝑚 · 1) = 𝑚) |
73 | 72 | adantl 275 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (𝑚 · 1) = 𝑚) |
74 | | mulcl 7859 |
. . . . 5
⊢ ((𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑚 · 𝑥) ∈ ℂ) |
75 | 74 | adantl 275 |
. . . 4
⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑚 · 𝑥) ∈ ℂ) |
76 | | 1cnd 7894 |
. . . 4
⊢ (𝜑 → 1 ∈
ℂ) |
77 | 26, 16 | eleqtrrd 2237 |
. . . 4
⊢ (𝜑 → 𝑁 ∈ (1...(♯‘𝐴))) |
78 | | eluzelz 9448 |
. . . . . 6
⊢ (𝑚 ∈
(ℤ≥‘𝑀) → 𝑚 ∈ ℤ) |
79 | | simpr 109 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐴) |
80 | 2 | ralrimiva 2530 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
81 | 80 | ad2antrr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) ∧ 𝑚 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
82 | | nfcsb1v 3064 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 |
83 | 82 | nfel1 2310 |
. . . . . . . . 9
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ∈ ℂ |
84 | | csbeq1a 3040 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵) |
85 | 84 | eleq1d 2226 |
. . . . . . . . 9
⊢ (𝑘 = 𝑚 → (𝐵 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌𝐵 ∈ ℂ)) |
86 | 83, 85 | rspc 2810 |
. . . . . . . 8
⊢ (𝑚 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑚 / 𝑘⦌𝐵 ∈ ℂ)) |
87 | 79, 81, 86 | sylc 62 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) ∧ 𝑚 ∈ 𝐴) → ⦋𝑚 / 𝑘⦌𝐵 ∈ ℂ) |
88 | | 1cnd 7894 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑚 ∈ 𝐴) → 1 ∈ ℂ) |
89 | | eleq1 2220 |
. . . . . . . . 9
⊢ (𝑘 = 𝑚 → (𝑘 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴)) |
90 | 89 | dcbid 824 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → (DECID 𝑘 ∈ 𝐴 ↔ DECID 𝑚 ∈ 𝐴)) |
91 | 3 | ralrimiva 2530 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ (ℤ≥‘𝑀)DECID 𝑘 ∈ 𝐴) |
92 | 91 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ∀𝑘 ∈
(ℤ≥‘𝑀)DECID 𝑘 ∈ 𝐴) |
93 | | simpr 109 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → 𝑚 ∈ (ℤ≥‘𝑀)) |
94 | 90, 92, 93 | rspcdva 2821 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → DECID
𝑚 ∈ 𝐴) |
95 | 87, 88, 94 | ifcldadc 3534 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) ∈ ℂ) |
96 | | nfcv 2299 |
. . . . . . 7
⊢
Ⅎ𝑘𝑚 |
97 | | nfv 1508 |
. . . . . . . 8
⊢
Ⅎ𝑘 𝑚 ∈ 𝐴 |
98 | | nfcv 2299 |
. . . . . . . 8
⊢
Ⅎ𝑘1 |
99 | 97, 82, 98 | nfif 3533 |
. . . . . . 7
⊢
Ⅎ𝑘if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) |
100 | 89, 84 | ifbieq1d 3527 |
. . . . . . 7
⊢ (𝑘 = 𝑚 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1)) |
101 | 96, 99, 100, 1 | fvmptf 5560 |
. . . . . 6
⊢ ((𝑚 ∈ ℤ ∧ if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) ∈ ℂ) → (𝐹‘𝑚) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1)) |
102 | 78, 95, 101 | syl2an2 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑚) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1)) |
103 | 102, 95 | eqeltrd 2234 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑚) ∈ ℂ) |
104 | | prodmodclem2.4 |
. . . . . 6
⊢ 𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1)) |
105 | | breq1 3968 |
. . . . . . 7
⊢ (𝑗 = 𝑚 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑚 ≤ (♯‘𝐴))) |
106 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑗 = 𝑚 → (𝐾‘𝑗) = (𝐾‘𝑚)) |
107 | 106 | csbeq1d 3038 |
. . . . . . 7
⊢ (𝑗 = 𝑚 → ⦋(𝐾‘𝑗) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵) |
108 | 105, 107 | ifbieq1d 3527 |
. . . . . 6
⊢ (𝑗 = 𝑚 → if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1) = if(𝑚 ≤ (♯‘𝐴), ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 1)) |
109 | | elnnuz 9475 |
. . . . . . . 8
⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈
(ℤ≥‘1)) |
110 | 109 | biimpri 132 |
. . . . . . 7
⊢ (𝑚 ∈
(ℤ≥‘1) → 𝑚 ∈ ℕ) |
111 | 110 | adantl 275 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ 𝑚 ∈
ℕ) |
112 | 22 | ad2antrr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
𝐾:(1...𝑁)⟶𝐴) |
113 | | 1zzd 9194 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) → 1
∈ ℤ) |
114 | 8 | ad2antrr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
𝑁 ∈
ℤ) |
115 | | eluzelz 9448 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈
(ℤ≥‘1) → 𝑚 ∈ ℤ) |
116 | 115 | ad2antlr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
𝑚 ∈
ℤ) |
117 | 113, 114,
116 | 3jca 1162 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
(1 ∈ ℤ ∧ 𝑁
∈ ℤ ∧ 𝑚
∈ ℤ)) |
118 | | eluzle 9451 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈
(ℤ≥‘1) → 1 ≤ 𝑚) |
119 | 118 | ad2antlr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) → 1
≤ 𝑚) |
120 | | simpr 109 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
𝑚 ≤ (♯‘𝐴)) |
121 | 15 | ad2antrr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
(♯‘𝐴) = 𝑁) |
122 | 120, 121 | breqtrd 3990 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
𝑚 ≤ 𝑁) |
123 | 119, 122 | jca 304 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
(1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑁)) |
124 | | elfz2 9919 |
. . . . . . . . . 10
⊢ (𝑚 ∈ (1...𝑁) ↔ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (1 ≤
𝑚 ∧ 𝑚 ≤ 𝑁))) |
125 | 117, 123,
124 | sylanbrc 414 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
𝑚 ∈ (1...𝑁)) |
126 | 112, 125 | ffvelrnd 5603 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
(𝐾‘𝑚) ∈ 𝐴) |
127 | 80 | ad2antrr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
128 | | nfcsb1v 3064 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 |
129 | 128 | nfel1 2310 |
. . . . . . . . 9
⊢
Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ |
130 | | csbeq1a 3040 |
. . . . . . . . . 10
⊢ (𝑘 = (𝐾‘𝑚) → 𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵) |
131 | 130 | eleq1d 2226 |
. . . . . . . . 9
⊢ (𝑘 = (𝐾‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)) |
132 | 129, 131 | rspc 2810 |
. . . . . . . 8
⊢ ((𝐾‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)) |
133 | 126, 127,
132 | sylc 62 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ 𝑚 ≤
(♯‘𝐴)) →
⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ) |
134 | | 1cnd 7894 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
∧ ¬ 𝑚 ≤
(♯‘𝐴)) → 1
∈ ℂ) |
135 | 111 | nnzd 9285 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ 𝑚 ∈
ℤ) |
136 | 15, 8 | eqeltrd 2234 |
. . . . . . . . 9
⊢ (𝜑 → (♯‘𝐴) ∈
ℤ) |
137 | 136 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ (♯‘𝐴)
∈ ℤ) |
138 | | zdcle 9240 |
. . . . . . . 8
⊢ ((𝑚 ∈ ℤ ∧
(♯‘𝐴) ∈
ℤ) → DECID 𝑚 ≤ (♯‘𝐴)) |
139 | 135, 137,
138 | syl2anc 409 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ DECID 𝑚 ≤ (♯‘𝐴)) |
140 | 133, 134,
139 | ifcldadc 3534 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ if(𝑚 ≤
(♯‘𝐴),
⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 1) ∈ ℂ) |
141 | 104, 108,
111, 140 | fvmptd3 5561 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ (𝐻‘𝑚) = if(𝑚 ≤ (♯‘𝐴), ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 1)) |
142 | 141, 140 | eqeltrd 2234 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1))
→ (𝐻‘𝑚) ∈
ℂ) |
143 | | eldifi 3229 |
. . . . . . 7
⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑚 ∈ (𝑀...(𝐾‘(♯‘𝐴)))) |
144 | | elfzelz 9928 |
. . . . . . 7
⊢ (𝑚 ∈ (𝑀...(𝐾‘(♯‘𝐴))) → 𝑚 ∈ ℤ) |
145 | 143, 144 | syl 14 |
. . . . . 6
⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑚 ∈ ℤ) |
146 | | eldifn 3230 |
. . . . . . . . 9
⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → ¬ 𝑚 ∈ 𝐴) |
147 | 146 | iffalsed 3515 |
. . . . . . . 8
⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) = 1) |
148 | | ax-1cn 7825 |
. . . . . . . 8
⊢ 1 ∈
ℂ |
149 | 147, 148 | eqeltrdi 2248 |
. . . . . . 7
⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) ∈ ℂ) |
150 | 149 | adantl 275 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) ∈ ℂ) |
151 | 145, 150,
101 | syl2an2 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑚) = if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1)) |
152 | 147 | adantl 275 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → if(𝑚 ∈ 𝐴, ⦋𝑚 / 𝑘⦌𝐵, 1) = 1) |
153 | 151, 152 | eqtrd 2190 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑚) = 1) |
154 | | elfzle2 9930 |
. . . . . . 7
⊢ (𝑥 ∈
(1...(♯‘𝐴))
→ 𝑥 ≤
(♯‘𝐴)) |
155 | 154 | adantl 275 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ≤ (♯‘𝐴)) |
156 | 155 | iftrued 3512 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ (♯‘𝐴), ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵) |
157 | | breq1 3968 |
. . . . . . 7
⊢ (𝑗 = 𝑥 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑥 ≤ (♯‘𝐴))) |
158 | | fveq2 5468 |
. . . . . . . 8
⊢ (𝑗 = 𝑥 → (𝐾‘𝑗) = (𝐾‘𝑥)) |
159 | 158 | csbeq1d 3038 |
. . . . . . 7
⊢ (𝑗 = 𝑥 → ⦋(𝐾‘𝑗) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵) |
160 | 157, 159 | ifbieq1d 3527 |
. . . . . 6
⊢ (𝑗 = 𝑥 → if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1) = if(𝑥 ≤ (♯‘𝐴), ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1)) |
161 | | elfznn 9956 |
. . . . . . 7
⊢ (𝑥 ∈
(1...(♯‘𝐴))
→ 𝑥 ∈
ℕ) |
162 | 161 | adantl 275 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ∈ ℕ) |
163 | 22 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝐾:(1...𝑁)⟶𝐴) |
164 | | simpr 109 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ∈ (1...(♯‘𝐴))) |
165 | 15 | adantr 274 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (♯‘𝐴) = 𝑁) |
166 | 165 | oveq2d 5840 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (1...(♯‘𝐴)) = (1...𝑁)) |
167 | 164, 166 | eleqtrd 2236 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ∈ (1...𝑁)) |
168 | 163, 167 | ffvelrnd 5603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ 𝐴) |
169 | 80 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
170 | | nfcsb1v 3064 |
. . . . . . . . . 10
⊢
Ⅎ𝑘⦋(𝐾‘𝑥) / 𝑘⦌𝐵 |
171 | 170 | nfel1 2310 |
. . . . . . . . 9
⊢
Ⅎ𝑘⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ |
172 | | csbeq1a 3040 |
. . . . . . . . . 10
⊢ (𝑘 = (𝐾‘𝑥) → 𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵) |
173 | 172 | eleq1d 2226 |
. . . . . . . . 9
⊢ (𝑘 = (𝐾‘𝑥) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ)) |
174 | 171, 173 | rspc 2810 |
. . . . . . . 8
⊢ ((𝐾‘𝑥) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ)) |
175 | 168, 169,
174 | sylc 62 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ) |
176 | 156, 175 | eqeltrd 2234 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ (♯‘𝐴), ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ) |
177 | 104, 160,
162, 176 | fvmptd3 5561 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = if(𝑥 ≤ (♯‘𝐴), ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1)) |
178 | 4 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑀)) |
179 | 178, 48 | sstrdi 3140 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ ℤ) |
180 | 179, 168 | sseldd 3129 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ ℤ) |
181 | 168 | iftrued 3512 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵) |
182 | 181, 175 | eqeltrd 2234 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ) |
183 | | nfcv 2299 |
. . . . . . . 8
⊢
Ⅎ𝑘(𝐾‘𝑥) |
184 | | nfv 1508 |
. . . . . . . . 9
⊢
Ⅎ𝑘(𝐾‘𝑥) ∈ 𝐴 |
185 | 184, 170,
98 | nfif 3533 |
. . . . . . . 8
⊢
Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) |
186 | | eleq1 2220 |
. . . . . . . . 9
⊢ (𝑘 = (𝐾‘𝑥) → (𝑘 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴)) |
187 | 186, 172 | ifbieq1d 3527 |
. . . . . . . 8
⊢ (𝑘 = (𝐾‘𝑥) → if(𝑘 ∈ 𝐴, 𝐵, 1) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1)) |
188 | 183, 185,
187, 1 | fvmptf 5560 |
. . . . . . 7
⊢ (((𝐾‘𝑥) ∈ ℤ ∧ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1)) |
189 | 180, 182,
188 | syl2anc 409 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1)) |
190 | 189, 181 | eqtrd 2190 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(𝐾‘𝑥)) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵) |
191 | 156, 177,
190 | 3eqtr4d 2200 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = (𝐹‘(𝐾‘𝑥))) |
192 | 71, 73, 75, 76, 5, 77, 4, 103, 142, 153, 191 | seq3coll 10713 |
. . 3
⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)) = (seq1( · , 𝐻)‘𝑁)) |
193 | | prodmodc.3 |
. . . 4
⊢ 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1)) |
194 | 7, 7 | jca 304 |
. . . 4
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ)) |
195 | 1, 2, 193, 104, 194, 10, 30 | prodmodclem3 11472 |
. . 3
⊢ (𝜑 → (seq1( · , 𝐺)‘𝑁) = (seq1( · , 𝐻)‘𝑁)) |
196 | 192, 195 | eqtr4d 2193 |
. 2
⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)) = (seq1( · , 𝐺)‘𝑁)) |
197 | 69, 196 | breqtrd 3990 |
1
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁)) |