Step | Hyp | Ref
| Expression |
1 | | fzfid 13546 |
. . . 4
⊢ (𝜑 → (1...𝑀) ∈ Fin) |
2 | | dvdsmulf1o.x |
. . . . 5
⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} |
3 | | dvdsmulf1o.1 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℕ) |
4 | | dvdsssfz1 15879 |
. . . . . 6
⊢ (𝑀 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ⊆ (1...𝑀)) |
5 | 3, 4 | syl 17 |
. . . . 5
⊢ (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ⊆ (1...𝑀)) |
6 | 2, 5 | eqsstrid 3949 |
. . . 4
⊢ (𝜑 → 𝑋 ⊆ (1...𝑀)) |
7 | 1, 6 | ssfid 8898 |
. . 3
⊢ (𝜑 → 𝑋 ∈ Fin) |
8 | | fzfid 13546 |
. . . . 5
⊢ (𝜑 → (1...𝑁) ∈ Fin) |
9 | | dvdsmulf1o.y |
. . . . . 6
⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} |
10 | | dvdsmulf1o.2 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℕ) |
11 | | dvdsssfz1 15879 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ (1...𝑁)) |
12 | 10, 11 | syl 17 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ (1...𝑁)) |
13 | 9, 12 | eqsstrid 3949 |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ (1...𝑁)) |
14 | 8, 13 | ssfid 8898 |
. . . 4
⊢ (𝜑 → 𝑌 ∈ Fin) |
15 | | fsumdvdsmul.5 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ) |
16 | 14, 15 | fsumcl 15297 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ 𝑌 𝐵 ∈ ℂ) |
17 | | fsumdvdsmul.4 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ) |
18 | 7, 16, 17 | fsummulc1 15349 |
. 2
⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵)) |
19 | 14 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑌 ∈ Fin) |
20 | 15 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ) |
21 | 19, 17, 20 | fsummulc2 15348 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑘 ∈ 𝑌 (𝐴 · 𝐵)) |
22 | | fsumdvdsmul.6 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) = 𝐷) |
23 | 22 | anassrs 471 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → (𝐴 · 𝐵) = 𝐷) |
24 | 23 | sumeq2dv 15267 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → Σ𝑘 ∈ 𝑌 (𝐴 · 𝐵) = Σ𝑘 ∈ 𝑌 𝐷) |
25 | 21, 24 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑘 ∈ 𝑌 𝐷) |
26 | 25 | sumeq2dv 15267 |
. 2
⊢ (𝜑 → Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷) |
27 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑧 = 〈𝑗, 𝑘〉 → ( · ‘𝑧) = ( ·
‘〈𝑗, 𝑘〉)) |
28 | | df-ov 7216 |
. . . . . . 7
⊢ (𝑗 · 𝑘) = ( · ‘〈𝑗, 𝑘〉) |
29 | 27, 28 | eqtr4di 2796 |
. . . . . 6
⊢ (𝑧 = 〈𝑗, 𝑘〉 → ( · ‘𝑧) = (𝑗 · 𝑘)) |
30 | 29 | csbeq1d 3815 |
. . . . 5
⊢ (𝑧 = 〈𝑗, 𝑘〉 → ⦋( ·
‘𝑧) / 𝑖⦌𝐶 = ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶) |
31 | | ovex 7246 |
. . . . . 6
⊢ (𝑗 · 𝑘) ∈ V |
32 | | fsumdvdsmul.7 |
. . . . . 6
⊢ (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷) |
33 | 31, 32 | csbie 3847 |
. . . . 5
⊢
⦋(𝑗
· 𝑘) / 𝑖⦌𝐶 = 𝐷 |
34 | 30, 33 | eqtrdi 2794 |
. . . 4
⊢ (𝑧 = 〈𝑗, 𝑘〉 → ⦋( ·
‘𝑧) / 𝑖⦌𝐶 = 𝐷) |
35 | 17 | adantrr 717 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐴 ∈ ℂ) |
36 | 15 | adantrl 716 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐵 ∈ ℂ) |
37 | 35, 36 | mulcld 10853 |
. . . . 5
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) ∈ ℂ) |
38 | 22, 37 | eqeltrrd 2839 |
. . . 4
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐷 ∈ ℂ) |
39 | 34, 7, 14, 38 | fsumxp 15336 |
. . 3
⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶) |
40 | | nfcv 2904 |
. . . . 5
⊢
Ⅎ𝑤𝐶 |
41 | | nfcsb1v 3836 |
. . . . 5
⊢
Ⅎ𝑖⦋𝑤 / 𝑖⦌𝐶 |
42 | | csbeq1a 3825 |
. . . . 5
⊢ (𝑖 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑖⦌𝐶) |
43 | 40, 41, 42 | cbvsumi 15261 |
. . . 4
⊢
Σ𝑖 ∈
𝑍 𝐶 = Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 |
44 | | csbeq1 3814 |
. . . . 5
⊢ (𝑤 = ( · ‘𝑧) → ⦋𝑤 / 𝑖⦌𝐶 = ⦋( · ‘𝑧) / 𝑖⦌𝐶) |
45 | | xpfi 8942 |
. . . . . 6
⊢ ((𝑋 ∈ Fin ∧ 𝑌 ∈ Fin) → (𝑋 × 𝑌) ∈ Fin) |
46 | 7, 14, 45 | syl2anc 587 |
. . . . 5
⊢ (𝜑 → (𝑋 × 𝑌) ∈ Fin) |
47 | | dvdsmulf1o.3 |
. . . . . 6
⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
48 | | dvdsmulf1o.z |
. . . . . 6
⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} |
49 | 3, 10, 47, 2, 9, 48 | dvdsmulf1o 26076 |
. . . . 5
⊢ (𝜑 → ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍) |
50 | | fvres 6736 |
. . . . . 6
⊢ (𝑧 ∈ (𝑋 × 𝑌) → (( · ↾ (𝑋 × 𝑌))‘𝑧) = ( · ‘𝑧)) |
51 | 50 | adantl 485 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (( · ↾ (𝑋 × 𝑌))‘𝑧) = ( · ‘𝑧)) |
52 | 38 | ralrimivva 3112 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 𝐷 ∈ ℂ) |
53 | 34 | eleq1d 2822 |
. . . . . . . . 9
⊢ (𝑧 = 〈𝑗, 𝑘〉 → (⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ 𝐷 ∈ ℂ)) |
54 | 53 | ralxp 5710 |
. . . . . . . 8
⊢
(∀𝑧 ∈
(𝑋 × 𝑌)⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 𝐷 ∈ ℂ) |
55 | 52, 54 | sylibr 237 |
. . . . . . 7
⊢ (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ) |
56 | | ax-mulf 10809 |
. . . . . . . . . 10
⊢ ·
:(ℂ × ℂ)⟶ℂ |
57 | | ffn 6545 |
. . . . . . . . . 10
⊢ (
· :(ℂ × ℂ)⟶ℂ → · Fn (ℂ
× ℂ)) |
58 | 56, 57 | ax-mp 5 |
. . . . . . . . 9
⊢ ·
Fn (ℂ × ℂ) |
59 | 2 | ssrab3 3995 |
. . . . . . . . . . 11
⊢ 𝑋 ⊆
ℕ |
60 | | nnsscn 11835 |
. . . . . . . . . . 11
⊢ ℕ
⊆ ℂ |
61 | 59, 60 | sstri 3910 |
. . . . . . . . . 10
⊢ 𝑋 ⊆
ℂ |
62 | 9 | ssrab3 3995 |
. . . . . . . . . . 11
⊢ 𝑌 ⊆
ℕ |
63 | 62, 60 | sstri 3910 |
. . . . . . . . . 10
⊢ 𝑌 ⊆
ℂ |
64 | | xpss12 5566 |
. . . . . . . . . 10
⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ ×
ℂ)) |
65 | 61, 63, 64 | mp2an 692 |
. . . . . . . . 9
⊢ (𝑋 × 𝑌) ⊆ (ℂ ×
ℂ) |
66 | 44 | eleq1d 2822 |
. . . . . . . . . 10
⊢ (𝑤 = ( · ‘𝑧) → (⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋(
· ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
67 | 66 | ralima 7054 |
. . . . . . . . 9
⊢ ((
· Fn (ℂ × ℂ) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) →
(∀𝑤 ∈ (
· “ (𝑋 ×
𝑌))⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
68 | 58, 65, 67 | mp2an 692 |
. . . . . . . 8
⊢
(∀𝑤 ∈ (
· “ (𝑋 ×
𝑌))⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ) |
69 | | df-ima 5564 |
. . . . . . . . . 10
⊢ (
· “ (𝑋 ×
𝑌)) = ran ( ·
↾ (𝑋 × 𝑌)) |
70 | | f1ofo 6668 |
. . . . . . . . . . 11
⊢ ((
· ↾ (𝑋 ×
𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍 → ( · ↾
(𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍) |
71 | | forn 6636 |
. . . . . . . . . . 11
⊢ ((
· ↾ (𝑋 ×
𝑌)):(𝑋 × 𝑌)–onto→𝑍 → ran ( · ↾ (𝑋 × 𝑌)) = 𝑍) |
72 | 49, 70, 71 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → ran ( · ↾
(𝑋 × 𝑌)) = 𝑍) |
73 | 69, 72 | syl5eq 2790 |
. . . . . . . . 9
⊢ (𝜑 → ( · “ (𝑋 × 𝑌)) = 𝑍) |
74 | 73 | raleqdv 3325 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑤 ∈ ( · “
(𝑋 × 𝑌))⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)) |
75 | 68, 74 | bitr3id 288 |
. . . . . . 7
⊢ (𝜑 → (∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)) |
76 | 55, 75 | mpbid 235 |
. . . . . 6
⊢ (𝜑 → ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ) |
77 | 76 | r19.21bi 3130 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ) |
78 | 44, 46, 49, 51, 77 | fsumf1o 15287 |
. . . 4
⊢ (𝜑 → Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶) |
79 | 43, 78 | syl5eq 2790 |
. . 3
⊢ (𝜑 → Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶) |
80 | 39, 79 | eqtr4d 2780 |
. 2
⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑖 ∈ 𝑍 𝐶) |
81 | 18, 26, 80 | 3eqtrd 2781 |
1
⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶) |