Step | Hyp | Ref
| Expression |
1 | | fzfid 13993 |
. . . 4
⊢ (𝜑 → (1...𝑀) ∈ Fin) |
2 | | mpodvdsmulf1o.x |
. . . . 5
⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} |
3 | | mpodvdsmulf1o.1 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℕ) |
4 | | dvdsssfz1 16320 |
. . . . . 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 4028 |
. . . 4
⊢ (𝜑 → 𝑋 ⊆ (1...𝑀)) |
7 | 1, 6 | ssfid 9301 |
. . 3
⊢ (𝜑 → 𝑋 ∈ Fin) |
8 | | fzfid 13993 |
. . . . 5
⊢ (𝜑 → (1...𝑁) ∈ Fin) |
9 | | mpodvdsmulf1o.y |
. . . . . 6
⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} |
10 | | mpodvdsmulf1o.2 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℕ) |
11 | | dvdsssfz1 16320 |
. . . . . . 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 4028 |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ (1...𝑁)) |
14 | 8, 13 | ssfid 9301 |
. . . 4
⊢ (𝜑 → 𝑌 ∈ Fin) |
15 | | fsumdvdsmul.5 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ) |
16 | 14, 15 | fsumcl 15737 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ 𝑌 𝐵 ∈ ℂ) |
17 | | fsumdvdsmul.4 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ) |
18 | 7, 16, 17 | fsummulc1 15789 |
. 2
⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵)) |
19 | 14 | adantr 479 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑌 ∈ Fin) |
20 | 15 | adantlr 713 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ) |
21 | 19, 17, 20 | fsummulc2 15788 |
. . . 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 466 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → (𝐴 · 𝐵) = 𝐷) |
24 | 23 | sumeq2dv 15707 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → Σ𝑘 ∈ 𝑌 (𝐴 · 𝐵) = Σ𝑘 ∈ 𝑌 𝐷) |
25 | 21, 24 | eqtrd 2766 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑘 ∈ 𝑌 𝐷) |
26 | 25 | sumeq2dv 15707 |
. 2
⊢ (𝜑 → Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷) |
27 | | elxpi 5704 |
. . . . . . 7
⊢ (𝑧 ∈ (𝑋 × 𝑌) → ∃𝑢∃𝑣(𝑧 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌))) |
28 | | fveq2 6901 |
. . . . . . . . . . . 12
⊢
(〈𝑢, 𝑣〉 = 𝑧 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧)) |
29 | 28 | eqcoms 2734 |
. . . . . . . . . . 11
⊢ (𝑧 = 〈𝑢, 𝑣〉 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧)) |
30 | | fveq2 6901 |
. . . . . . . . . . . 12
⊢
(〈𝑢, 𝑣〉 = 𝑧 → ( · ‘〈𝑢, 𝑣〉) = ( · ‘𝑧)) |
31 | 30 | eqcoms 2734 |
. . . . . . . . . . 11
⊢ (𝑧 = 〈𝑢, 𝑣〉 → ( · ‘〈𝑢, 𝑣〉) = ( · ‘𝑧)) |
32 | 29, 31 | eqeq12d 2742 |
. . . . . . . . . 10
⊢ (𝑧 = 〈𝑢, 𝑣〉 → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘〈𝑢, 𝑣〉) = ( · ‘〈𝑢, 𝑣〉) ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))) |
33 | 32 | biimpd 228 |
. . . . . . . . 9
⊢ (𝑧 = 〈𝑢, 𝑣〉 → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘〈𝑢, 𝑣〉) = ( · ‘〈𝑢, 𝑣〉) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))) |
34 | 2 | ssrab3 4079 |
. . . . . . . . . . . 12
⊢ 𝑋 ⊆
ℕ |
35 | | nnsscn 12269 |
. . . . . . . . . . . 12
⊢ ℕ
⊆ ℂ |
36 | 34, 35 | sstri 3989 |
. . . . . . . . . . 11
⊢ 𝑋 ⊆
ℂ |
37 | 36 | sseli 3975 |
. . . . . . . . . 10
⊢ (𝑢 ∈ 𝑋 → 𝑢 ∈ ℂ) |
38 | 9 | ssrab3 4079 |
. . . . . . . . . . . 12
⊢ 𝑌 ⊆
ℕ |
39 | 38, 35 | sstri 3989 |
. . . . . . . . . . 11
⊢ 𝑌 ⊆
ℂ |
40 | 39 | sseli 3975 |
. . . . . . . . . 10
⊢ (𝑣 ∈ 𝑌 → 𝑣 ∈ ℂ) |
41 | | ovmpot 7587 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = (𝑢 · 𝑣)) |
42 | | df-ov 7427 |
. . . . . . . . . . 11
⊢ (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘〈𝑢, 𝑣〉) |
43 | | df-ov 7427 |
. . . . . . . . . . 11
⊢ (𝑢 · 𝑣) = ( · ‘〈𝑢, 𝑣〉) |
44 | 41, 42, 43 | 3eqtr3g 2789 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘〈𝑢, 𝑣〉) = ( · ‘〈𝑢, 𝑣〉)) |
45 | 37, 40, 44 | syl2an 594 |
. . . . . . . . 9
⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘〈𝑢, 𝑣〉) = ( · ‘〈𝑢, 𝑣〉)) |
46 | 33, 45 | impel 504 |
. . . . . . . 8
⊢ ((𝑧 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)) |
47 | 46 | exlimivv 1928 |
. . . . . . 7
⊢
(∃𝑢∃𝑣(𝑧 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)) |
48 | 27, 47 | syl 17 |
. . . . . 6
⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)) |
49 | 48 | eqcomd 2732 |
. . . . 5
⊢ (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧)) |
50 | 49 | csbeq1d 3896 |
. . . 4
⊢ (𝑧 ∈ (𝑋 × 𝑌) → ⦋( ·
‘𝑧) / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶) |
51 | 50 | sumeq2i 15703 |
. . 3
⊢
Σ𝑧 ∈
(𝑋 × 𝑌)⦋( ·
‘𝑧) / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 |
52 | | fveq2 6901 |
. . . . . . 7
⊢ (𝑧 = 〈𝑗, 𝑘〉 → ( · ‘𝑧) = ( ·
‘〈𝑗, 𝑘〉)) |
53 | | df-ov 7427 |
. . . . . . 7
⊢ (𝑗 · 𝑘) = ( · ‘〈𝑗, 𝑘〉) |
54 | 52, 53 | eqtr4di 2784 |
. . . . . 6
⊢ (𝑧 = 〈𝑗, 𝑘〉 → ( · ‘𝑧) = (𝑗 · 𝑘)) |
55 | 54 | csbeq1d 3896 |
. . . . 5
⊢ (𝑧 = 〈𝑗, 𝑘〉 → ⦋( ·
‘𝑧) / 𝑖⦌𝐶 = ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶) |
56 | | ovex 7457 |
. . . . . 6
⊢ (𝑗 · 𝑘) ∈ V |
57 | | fsumdvdsmul.7 |
. . . . . 6
⊢ (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷) |
58 | 56, 57 | csbie 3928 |
. . . . 5
⊢
⦋(𝑗
· 𝑘) / 𝑖⦌𝐶 = 𝐷 |
59 | 55, 58 | eqtrdi 2782 |
. . . 4
⊢ (𝑧 = 〈𝑗, 𝑘〉 → ⦋( ·
‘𝑧) / 𝑖⦌𝐶 = 𝐷) |
60 | 17 | adantrr 715 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐴 ∈ ℂ) |
61 | 15 | adantrl 714 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐵 ∈ ℂ) |
62 | 60, 61 | mulcld 11284 |
. . . . 5
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) ∈ ℂ) |
63 | 22, 62 | eqeltrrd 2827 |
. . . 4
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐷 ∈ ℂ) |
64 | 59, 7, 14, 63 | fsumxp 15776 |
. . 3
⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶) |
65 | | nfcv 2892 |
. . . . 5
⊢
Ⅎ𝑤𝐶 |
66 | | nfcsb1v 3917 |
. . . . 5
⊢
Ⅎ𝑖⦋𝑤 / 𝑖⦌𝐶 |
67 | | csbeq1a 3906 |
. . . . 5
⊢ (𝑖 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑖⦌𝐶) |
68 | 65, 66, 67 | cbvsumi 15701 |
. . . 4
⊢
Σ𝑖 ∈
𝑍 𝐶 = Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 |
69 | | csbeq1 3895 |
. . . . 5
⊢ (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → ⦋𝑤 / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶) |
70 | | xpfi 9360 |
. . . . . 6
⊢ ((𝑋 ∈ Fin ∧ 𝑌 ∈ Fin) → (𝑋 × 𝑌) ∈ Fin) |
71 | 7, 14, 70 | syl2anc 582 |
. . . . 5
⊢ (𝜑 → (𝑋 × 𝑌) ∈ Fin) |
72 | | mpodvdsmulf1o.3 |
. . . . . 6
⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
73 | | mpodvdsmulf1o.z |
. . . . . 6
⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} |
74 | 3, 10, 72, 2, 9, 73 | mpodvdsmulf1o 27222 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍) |
75 | | fvres 6920 |
. . . . . 6
⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧)) |
76 | 75 | adantl 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧)) |
77 | 63 | ralrimivva 3191 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 𝐷 ∈ ℂ) |
78 | 59 | eleq1d 2811 |
. . . . . . . . . 10
⊢ (𝑧 = 〈𝑗, 𝑘〉 → (⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ 𝐷 ∈ ℂ)) |
79 | 78 | ralxp 5848 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
(𝑋 × 𝑌)⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 𝐷 ∈ ℂ) |
80 | 77, 79 | sylibr 233 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ) |
81 | | fveq2 6901 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑤 → ( · ‘𝑧) = ( · ‘𝑤)) |
82 | 81 | csbeq1d 3896 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑤 → ⦋( ·
‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶) |
83 | 82 | eleq1d 2811 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑤 → (⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋(
· ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ)) |
84 | 83 | cbvralvw 3225 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
(𝑋 × 𝑌)⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑤 ∈ (𝑋 × 𝑌)⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ) |
85 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)) |
86 | 82 | eqcoms 2734 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑧 → ⦋( ·
‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶) |
87 | 86 | adantl 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → ⦋( ·
‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶) |
88 | 87 | eleq1d 2811 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋(
· ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ)) |
89 | 50 | eleq1d 2811 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ (𝑋 × 𝑌) → (⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔
⦋((𝑥 ∈
ℂ, 𝑦 ∈ ℂ
↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
90 | 89 | adantr 479 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔
⦋((𝑥 ∈
ℂ, 𝑦 ∈ ℂ
↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
91 | 88, 90 | bitr3d 280 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( ·
‘𝑤) / 𝑖⦌𝐶 ∈ ℂ ↔
⦋((𝑥 ∈
ℂ, 𝑦 ∈ ℂ
↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
92 | 85, 91 | rspcdv 3600 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ (𝑋 × 𝑌) → (∀𝑤 ∈ (𝑋 × 𝑌)⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ →
⦋((𝑥 ∈
ℂ, 𝑦 ∈ ℂ
↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
93 | 92 | com12 32 |
. . . . . . . . . 10
⊢
(∀𝑤 ∈
(𝑋 × 𝑌)⦋( ·
‘𝑤) / 𝑖⦌𝐶 ∈ ℂ → (𝑧 ∈ (𝑋 × 𝑌) → ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
94 | 93 | ralrimiv 3135 |
. . . . . . . . 9
⊢
(∀𝑤 ∈
(𝑋 × 𝑌)⦋( ·
‘𝑤) / 𝑖⦌𝐶 ∈ ℂ → ∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ) |
95 | 84, 94 | sylbi 216 |
. . . . . . . 8
⊢
(∀𝑧 ∈
(𝑋 × 𝑌)⦋( ·
‘𝑧) / 𝑖⦌𝐶 ∈ ℂ → ∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ) |
96 | 80, 95 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ) |
97 | | mpomulf 11253 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ ×
ℂ)⟶ℂ |
98 | | ffn 6728 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
→ (𝑥 ∈ ℂ,
𝑦 ∈ ℂ ↦
(𝑥 · 𝑦)) Fn (ℂ ×
ℂ)) |
99 | 97, 98 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ ×
ℂ) |
100 | | xpss12 5697 |
. . . . . . . . . 10
⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ ×
ℂ)) |
101 | 36, 39, 100 | mp2an 690 |
. . . . . . . . 9
⊢ (𝑋 × 𝑌) ⊆ (ℂ ×
ℂ) |
102 | 69 | eleq1d 2811 |
. . . . . . . . . 10
⊢ (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → (⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔
⦋((𝑥 ∈
ℂ, 𝑦 ∈ ℂ
↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
103 | 102 | ralima 7255 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) →
(∀𝑤 ∈ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌))⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)) |
104 | 99, 101, 103 | mp2an 690 |
. . . . . . . 8
⊢
(∀𝑤 ∈
((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌))⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ) |
105 | | df-ima 5695 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) |
106 | | f1ofo 6850 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍) |
107 | | forn 6818 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍 → ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) = 𝑍) |
108 | 74, 106, 107 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) = 𝑍) |
109 | 105, 108 | eqtrid 2778 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = 𝑍) |
110 | 109 | raleqdv 3315 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑤 ∈ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌))⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)) |
111 | 104, 110 | bitr3id 284 |
. . . . . . 7
⊢ (𝜑 → (∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)) |
112 | 96, 111 | mpbid 231 |
. . . . . 6
⊢ (𝜑 → ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ) |
113 | 112 | r19.21bi 3239 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ) |
114 | 69, 71, 74, 76, 113 | fsumf1o 15727 |
. . . 4
⊢ (𝜑 → Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶) |
115 | 68, 114 | eqtrid 2778 |
. . 3
⊢ (𝜑 → Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶) |
116 | 51, 64, 115 | 3eqtr4a 2792 |
. 2
⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑖 ∈ 𝑍 𝐶) |
117 | 18, 26, 116 | 3eqtrd 2770 |
1
⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶) |