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Theorem fsumdvdsmul 27223
Description: Product of two divisor sums. (This is also the main part of the proof that "Σ𝑘𝑁𝐹(𝑘) is a multiplicative function if 𝐹 is".) (Contributed by Mario Carneiro, 2-Jul-2015.) Avoid ax-mulf 11238. (Revised by GG, 18-Apr-2025.)
Hypotheses
Ref Expression
mpodvdsmulf1o.1 (𝜑𝑀 ∈ ℕ)
mpodvdsmulf1o.2 (𝜑𝑁 ∈ ℕ)
mpodvdsmulf1o.3 (𝜑 → (𝑀 gcd 𝑁) = 1)
mpodvdsmulf1o.x 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}
mpodvdsmulf1o.y 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}
mpodvdsmulf1o.z 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}
fsumdvdsmul.4 ((𝜑𝑗𝑋) → 𝐴 ∈ ℂ)
fsumdvdsmul.5 ((𝜑𝑘𝑌) → 𝐵 ∈ ℂ)
fsumdvdsmul.6 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → (𝐴 · 𝐵) = 𝐷)
fsumdvdsmul.7 (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)
Assertion
Ref Expression
fsumdvdsmul (𝜑 → (Σ𝑗𝑋 𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑖𝑍 𝐶)
Distinct variable groups:   𝑥,𝑖,𝑗,𝑘   𝑥,𝑀   𝑥,𝑁   𝑖,𝑋,𝑗   𝑖,𝑌,𝑗   𝑖,𝑍,𝑗   𝜑,𝑖,𝑗   𝑘,𝑋   𝑘,𝑌   𝐴,𝑘   𝐵,𝑗   𝐶,𝑗,𝑘   𝐷,𝑖   𝜑,𝑘
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑖,𝑗)   𝐵(𝑥,𝑖,𝑘)   𝐶(𝑥,𝑖)   𝐷(𝑥,𝑗,𝑘)   𝑀(𝑖,𝑗,𝑘)   𝑁(𝑖,𝑗,𝑘)   𝑋(𝑥)   𝑌(𝑥)   𝑍(𝑥,𝑘)

Proof of Theorem fsumdvdsmul
Dummy variables 𝑦 𝑧 𝑤 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 13993 . . . 4 (𝜑 → (1...𝑀) ∈ Fin)
2 mpodvdsmulf1o.x . . . . 5 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}
3 mpodvdsmulf1o.1 . . . . . 6 (𝜑𝑀 ∈ ℕ)
4 dvdsssfz1 16320 . . . . . 6 (𝑀 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑀} ⊆ (1...𝑀))
53, 4syl 17 . . . . 5 (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥𝑀} ⊆ (1...𝑀))
62, 5eqsstrid 4028 . . . 4 (𝜑𝑋 ⊆ (1...𝑀))
71, 6ssfid 9301 . . 3 (𝜑𝑋 ∈ Fin)
8 fzfid 13993 . . . . 5 (𝜑 → (1...𝑁) ∈ Fin)
9 mpodvdsmulf1o.y . . . . . 6 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}
10 mpodvdsmulf1o.2 . . . . . . 7 (𝜑𝑁 ∈ ℕ)
11 dvdsssfz1 16320 . . . . . . 7 (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑁} ⊆ (1...𝑁))
1210, 11syl 17 . . . . . 6 (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥𝑁} ⊆ (1...𝑁))
139, 12eqsstrid 4028 . . . . 5 (𝜑𝑌 ⊆ (1...𝑁))
148, 13ssfid 9301 . . . 4 (𝜑𝑌 ∈ Fin)
15 fsumdvdsmul.5 . . . 4 ((𝜑𝑘𝑌) → 𝐵 ∈ ℂ)
1614, 15fsumcl 15737 . . 3 (𝜑 → Σ𝑘𝑌 𝐵 ∈ ℂ)
17 fsumdvdsmul.4 . . 3 ((𝜑𝑗𝑋) → 𝐴 ∈ ℂ)
187, 16, 17fsummulc1 15789 . 2 (𝜑 → (Σ𝑗𝑋 𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑗𝑋 (𝐴 · Σ𝑘𝑌 𝐵))
1914adantr 479 . . . . 5 ((𝜑𝑗𝑋) → 𝑌 ∈ Fin)
2015adantlr 713 . . . . 5 (((𝜑𝑗𝑋) ∧ 𝑘𝑌) → 𝐵 ∈ ℂ)
2119, 17, 20fsummulc2 15788 . . . 4 ((𝜑𝑗𝑋) → (𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑘𝑌 (𝐴 · 𝐵))
22 fsumdvdsmul.6 . . . . . 6 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → (𝐴 · 𝐵) = 𝐷)
2322anassrs 466 . . . . 5 (((𝜑𝑗𝑋) ∧ 𝑘𝑌) → (𝐴 · 𝐵) = 𝐷)
2423sumeq2dv 15707 . . . 4 ((𝜑𝑗𝑋) → Σ𝑘𝑌 (𝐴 · 𝐵) = Σ𝑘𝑌 𝐷)
2521, 24eqtrd 2766 . . 3 ((𝜑𝑗𝑋) → (𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑘𝑌 𝐷)
2625sumeq2dv 15707 . 2 (𝜑 → Σ𝑗𝑋 (𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑗𝑋 Σ𝑘𝑌 𝐷)
27 elxpi 5704 . . . . . . 7 (𝑧 ∈ (𝑋 × 𝑌) → ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢𝑋𝑣𝑌)))
28 fveq2 6901 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ = 𝑧 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
2928eqcoms 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))
3130eqcoms 2734 . . . . . . . . . . 11 (𝑧 = ⟨𝑢, 𝑣⟩ → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
3229, 31eqeq12d 2742 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩) ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)))
3332biimpd 228 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)))
342ssrab3 4079 . . . . . . . . . . . 12 𝑋 ⊆ ℕ
35 nnsscn 12269 . . . . . . . . . . . 12 ℕ ⊆ ℂ
3634, 35sstri 3989 . . . . . . . . . . 11 𝑋 ⊆ ℂ
3736sseli 3975 . . . . . . . . . 10 (𝑢𝑋𝑢 ∈ ℂ)
389ssrab3 4079 . . . . . . . . . . . 12 𝑌 ⊆ ℕ
3938, 35sstri 3989 . . . . . . . . . . 11 𝑌 ⊆ ℂ
4039sseli 3975 . . . . . . . . . 10 (𝑣𝑌𝑣 ∈ 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⟩)
4441, 42, 433eqtr3g 2789 . . . . . . . . . 10 ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
4537, 40, 44syl2an 594 . . . . . . . . 9 ((𝑢𝑋𝑣𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
4633, 45impel 504 . . . . . . . 8 ((𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢𝑋𝑣𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
4746exlimivv 1928 . . . . . . 7 (∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢𝑋𝑣𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
4827, 47syl 17 . . . . . 6 (𝑧 ∈ (𝑋 × 𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
4948eqcomd 2732 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
5049csbeq1d 3896 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) / 𝑖𝐶 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶)
5150sumeq2i 15703 . . 3 Σ𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶
52 fveq2 6901 . . . . . . 7 (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = ( · ‘⟨𝑗, 𝑘⟩))
53 df-ov 7427 . . . . . . 7 (𝑗 · 𝑘) = ( · ‘⟨𝑗, 𝑘⟩)
5452, 53eqtr4di 2784 . . . . . 6 (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = (𝑗 · 𝑘))
5554csbeq1d 3896 . . . . 5 (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) / 𝑖𝐶 = (𝑗 · 𝑘) / 𝑖𝐶)
56 ovex 7457 . . . . . 6 (𝑗 · 𝑘) ∈ V
57 fsumdvdsmul.7 . . . . . 6 (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)
5856, 57csbie 3928 . . . . 5 (𝑗 · 𝑘) / 𝑖𝐶 = 𝐷
5955, 58eqtrdi 2782 . . . 4 (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) / 𝑖𝐶 = 𝐷)
6017adantrr 715 . . . . . 6 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → 𝐴 ∈ ℂ)
6115adantrl 714 . . . . . 6 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → 𝐵 ∈ ℂ)
6260, 61mulcld 11284 . . . . 5 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → (𝐴 · 𝐵) ∈ ℂ)
6322, 62eqeltrrd 2827 . . . 4 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → 𝐷 ∈ ℂ)
6459, 7, 14, 63fsumxp 15776 . . 3 (𝜑 → Σ𝑗𝑋 Σ𝑘𝑌 𝐷 = Σ𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶)
65 nfcv 2892 . . . . 5 𝑤𝐶
66 nfcsb1v 3917 . . . . 5 𝑖𝑤 / 𝑖𝐶
67 csbeq1a 3906 . . . . 5 (𝑖 = 𝑤𝐶 = 𝑤 / 𝑖𝐶)
6865, 66, 67cbvsumi 15701 . . . 4 Σ𝑖𝑍 𝐶 = Σ𝑤𝑍 𝑤 / 𝑖𝐶
69 csbeq1 3895 . . . . 5 (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → 𝑤 / 𝑖𝐶 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶)
70 xpfi 9360 . . . . . 6 ((𝑋 ∈ Fin ∧ 𝑌 ∈ Fin) → (𝑋 × 𝑌) ∈ Fin)
717, 14, 70syl2anc 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)}
743, 10, 72, 2, 9, 73mpodvdsmulf1o 27222 . . . . 5 (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto𝑍)
75 fvres 6920 . . . . . 6 (𝑧 ∈ (𝑋 × 𝑌) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
7675adantl 480 . . . . 5 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
7763ralrimivva 3191 . . . . . . . . 9 (𝜑 → ∀𝑗𝑋𝑘𝑌 𝐷 ∈ ℂ)
7859eleq1d 2811 . . . . . . . . . 10 (𝑧 = ⟨𝑗, 𝑘⟩ → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ 𝐷 ∈ ℂ))
7978ralxp 5848 . . . . . . . . 9 (∀𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ∀𝑗𝑋𝑘𝑌 𝐷 ∈ ℂ)
8077, 79sylibr 233 . . . . . . . 8 (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 ∈ ℂ)
81 fveq2 6901 . . . . . . . . . . . 12 (𝑧 = 𝑤 → ( · ‘𝑧) = ( · ‘𝑤))
8281csbeq1d 3896 . . . . . . . . . . 11 (𝑧 = 𝑤( · ‘𝑧) / 𝑖𝐶 = ( · ‘𝑤) / 𝑖𝐶)
8382eleq1d 2811 . . . . . . . . . 10 (𝑧 = 𝑤 → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ( · ‘𝑤) / 𝑖𝐶 ∈ ℂ))
8483cbvralvw 3225 . . . . . . . . 9 (∀𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ∀𝑤 ∈ (𝑋 × 𝑌)( · ‘𝑤) / 𝑖𝐶 ∈ ℂ)
85 id 22 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌))
8682eqcoms 2734 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧( · ‘𝑧) / 𝑖𝐶 = ( · ‘𝑤) / 𝑖𝐶)
8786adantl 480 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → ( · ‘𝑧) / 𝑖𝐶 = ( · ‘𝑤) / 𝑖𝐶)
8887eleq1d 2811 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ( · ‘𝑤) / 𝑖𝐶 ∈ ℂ))
8950eleq1d 2811 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋 × 𝑌) → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
9089adantr 479 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
9188, 90bitr3d 280 . . . . . . . . . . . 12 ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (( · ‘𝑤) / 𝑖𝐶 ∈ ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
9285, 91rspcdv 3600 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 × 𝑌) → (∀𝑤 ∈ (𝑋 × 𝑌)( · ‘𝑤) / 𝑖𝐶 ∈ ℂ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
9392com12 32 . . . . . . . . . 10 (∀𝑤 ∈ (𝑋 × 𝑌)( · ‘𝑤) / 𝑖𝐶 ∈ ℂ → (𝑧 ∈ (𝑋 × 𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
9493ralrimiv 3135 . . . . . . . . 9 (∀𝑤 ∈ (𝑋 × 𝑌)( · ‘𝑤) / 𝑖𝐶 ∈ ℂ → ∀𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ)
9584, 94sylbi 216 . . . . . . . 8 (∀𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 ∈ ℂ → ∀𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ)
9680, 95syl 17 . . . . . . 7 (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ 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))
9997, 98ax-mp 5 . . . . . . . . 9 (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ)
100 xpss12 5697 . . . . . . . . . 10 ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
10136, 39, 100mp2an 690 . . . . . . . . 9 (𝑋 × 𝑌) ⊆ (ℂ × ℂ)
10269eleq1d 2811 . . . . . . . . . 10 (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → (𝑤 / 𝑖𝐶 ∈ ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
103102ralima 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 / 𝑖𝐶 ∈ ℂ ↔ ∀𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
10499, 101, 103mp2an 690 . . . . . . . 8 (∀𝑤 ∈ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌))𝑤 / 𝑖𝐶 ∈ ℂ ↔ ∀𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ 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-onto𝑍 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑍)
107 forn 6818 . . . . . . . . . . 11 (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑍 → ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) = 𝑍)
10874, 106, 1073syl 18 . . . . . . . . . 10 (𝜑 → ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) = 𝑍)
109105, 108eqtrid 2778 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = 𝑍)
110109raleqdv 3315 . . . . . . . 8 (𝜑 → (∀𝑤 ∈ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌))𝑤 / 𝑖𝐶 ∈ ℂ ↔ ∀𝑤𝑍 𝑤 / 𝑖𝐶 ∈ ℂ))
111104, 110bitr3id 284 . . . . . . 7 (𝜑 → (∀𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ∀𝑤𝑍 𝑤 / 𝑖𝐶 ∈ ℂ))
11296, 111mpbid 231 . . . . . 6 (𝜑 → ∀𝑤𝑍 𝑤 / 𝑖𝐶 ∈ ℂ)
113112r19.21bi 3239 . . . . 5 ((𝜑𝑤𝑍) → 𝑤 / 𝑖𝐶 ∈ ℂ)
11469, 71, 74, 76, 113fsumf1o 15727 . . . 4 (𝜑 → Σ𝑤𝑍 𝑤 / 𝑖𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶)
11568, 114eqtrid 2778 . . 3 (𝜑 → Σ𝑖𝑍 𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶)
11651, 64, 1153eqtr4a 2792 . 2 (𝜑 → Σ𝑗𝑋 Σ𝑘𝑌 𝐷 = Σ𝑖𝑍 𝐶)
11718, 26, 1163eqtrd 2770 1 (𝜑 → (Σ𝑗𝑋 𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑖𝑍 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  wral 3051  {crab 3419  csb 3892  wss 3947  cop 4639   class class class wbr 5153   × cxp 5680  ran crn 5683  cres 5684  cima 5685   Fn wfn 6549  wf 6550  ontowfo 6552  1-1-ontowf1o 6553  cfv 6554  (class class class)co 7424  cmpo 7426  Fincfn 8974  cc 11156  1c1 11159   · cmul 11163  cn 12264  ...cfz 13538  Σcsu 15690  cdvds 16256   gcd cgcd 16494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-fz 13539  df-fzo 13682  df-fl 13812  df-mod 13890  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-sum 15691  df-dvds 16257  df-gcd 16495
This theorem is referenced by:  sgmmul  27230  dchrisum0fmul  27535
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