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Theorem fsumdvdsmul 27252
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 11232. (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 14010 . . . 4 (𝜑 → (1...𝑀) ∈ Fin)
2 mpodvdsmulf1o.x . . . . 5 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥𝑀}
3 mpodvdsmulf1o.1 . . . . . 6 (𝜑𝑀 ∈ ℕ)
4 dvdsssfz1 16351 . . . . . 6 (𝑀 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑀} ⊆ (1...𝑀))
53, 4syl 17 . . . . 5 (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥𝑀} ⊆ (1...𝑀))
62, 5eqsstrid 4043 . . . 4 (𝜑𝑋 ⊆ (1...𝑀))
71, 6ssfid 9298 . . 3 (𝜑𝑋 ∈ Fin)
8 fzfid 14010 . . . . 5 (𝜑 → (1...𝑁) ∈ Fin)
9 mpodvdsmulf1o.y . . . . . 6 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}
10 mpodvdsmulf1o.2 . . . . . . 7 (𝜑𝑁 ∈ ℕ)
11 dvdsssfz1 16351 . . . . . . 7 (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑁} ⊆ (1...𝑁))
1210, 11syl 17 . . . . . 6 (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥𝑁} ⊆ (1...𝑁))
139, 12eqsstrid 4043 . . . . 5 (𝜑𝑌 ⊆ (1...𝑁))
148, 13ssfid 9298 . . . 4 (𝜑𝑌 ∈ Fin)
15 fsumdvdsmul.5 . . . 4 ((𝜑𝑘𝑌) → 𝐵 ∈ ℂ)
1614, 15fsumcl 15765 . . 3 (𝜑 → Σ𝑘𝑌 𝐵 ∈ ℂ)
17 fsumdvdsmul.4 . . 3 ((𝜑𝑗𝑋) → 𝐴 ∈ ℂ)
187, 16, 17fsummulc1 15817 . 2 (𝜑 → (Σ𝑗𝑋 𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑗𝑋 (𝐴 · Σ𝑘𝑌 𝐵))
1914adantr 480 . . . . 5 ((𝜑𝑗𝑋) → 𝑌 ∈ Fin)
2015adantlr 715 . . . . 5 (((𝜑𝑗𝑋) ∧ 𝑘𝑌) → 𝐵 ∈ ℂ)
2119, 17, 20fsummulc2 15816 . . . 4 ((𝜑𝑗𝑋) → (𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑘𝑌 (𝐴 · 𝐵))
22 fsumdvdsmul.6 . . . . . 6 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → (𝐴 · 𝐵) = 𝐷)
2322anassrs 467 . . . . 5 (((𝜑𝑗𝑋) ∧ 𝑘𝑌) → (𝐴 · 𝐵) = 𝐷)
2423sumeq2dv 15734 . . . 4 ((𝜑𝑗𝑋) → Σ𝑘𝑌 (𝐴 · 𝐵) = Σ𝑘𝑌 𝐷)
2521, 24eqtrd 2774 . . 3 ((𝜑𝑗𝑋) → (𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑘𝑌 𝐷)
2625sumeq2dv 15734 . 2 (𝜑 → Σ𝑗𝑋 (𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑗𝑋 Σ𝑘𝑌 𝐷)
27 elxpi 5710 . . . . . . 7 (𝑧 ∈ (𝑋 × 𝑌) → ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢𝑋𝑣𝑌)))
28 fveq2 6906 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ = 𝑧 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
2928eqcoms 2742 . . . . . . . . . . 11 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
30 fveq2 6906 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ = 𝑧 → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
3130eqcoms 2742 . . . . . . . . . . 11 (𝑧 = ⟨𝑢, 𝑣⟩ → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
3229, 31eqeq12d 2750 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩) ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)))
3332biimpd 229 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)))
342ssrab3 4091 . . . . . . . . . . . 12 𝑋 ⊆ ℕ
35 nnsscn 12268 . . . . . . . . . . . 12 ℕ ⊆ ℂ
3634, 35sstri 4004 . . . . . . . . . . 11 𝑋 ⊆ ℂ
3736sseli 3990 . . . . . . . . . 10 (𝑢𝑋𝑢 ∈ ℂ)
389ssrab3 4091 . . . . . . . . . . . 12 𝑌 ⊆ ℕ
3938, 35sstri 4004 . . . . . . . . . . 11 𝑌 ⊆ ℂ
4039sseli 3990 . . . . . . . . . 10 (𝑣𝑌𝑣 ∈ ℂ)
41 ovmpot 7593 . . . . . . . . . . 11 ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = (𝑢 · 𝑣))
42 df-ov 7433 . . . . . . . . . . 11 (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩)
43 df-ov 7433 . . . . . . . . . . 11 (𝑢 · 𝑣) = ( · ‘⟨𝑢, 𝑣⟩)
4441, 42, 433eqtr3g 2797 . . . . . . . . . 10 ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
4537, 40, 44syl2an 596 . . . . . . . . 9 ((𝑢𝑋𝑣𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
4633, 45impel 505 . . . . . . . 8 ((𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢𝑋𝑣𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
4746exlimivv 1929 . . . . . . 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 2740 . . . . 5 (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
5049csbeq1d 3911 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) / 𝑖𝐶 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶)
5150sumeq2i 15730 . . 3 Σ𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶
52 fveq2 6906 . . . . . . 7 (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = ( · ‘⟨𝑗, 𝑘⟩))
53 df-ov 7433 . . . . . . 7 (𝑗 · 𝑘) = ( · ‘⟨𝑗, 𝑘⟩)
5452, 53eqtr4di 2792 . . . . . 6 (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = (𝑗 · 𝑘))
5554csbeq1d 3911 . . . . 5 (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) / 𝑖𝐶 = (𝑗 · 𝑘) / 𝑖𝐶)
56 ovex 7463 . . . . . 6 (𝑗 · 𝑘) ∈ V
57 fsumdvdsmul.7 . . . . . 6 (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)
5856, 57csbie 3943 . . . . 5 (𝑗 · 𝑘) / 𝑖𝐶 = 𝐷
5955, 58eqtrdi 2790 . . . 4 (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) / 𝑖𝐶 = 𝐷)
6017adantrr 717 . . . . . 6 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → 𝐴 ∈ ℂ)
6115adantrl 716 . . . . . 6 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → 𝐵 ∈ ℂ)
6260, 61mulcld 11278 . . . . 5 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → (𝐴 · 𝐵) ∈ ℂ)
6322, 62eqeltrrd 2839 . . . 4 ((𝜑 ∧ (𝑗𝑋𝑘𝑌)) → 𝐷 ∈ ℂ)
6459, 7, 14, 63fsumxp 15804 . . 3 (𝜑 → Σ𝑗𝑋 Σ𝑘𝑌 𝐷 = Σ𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶)
65 csbeq1a 3921 . . . . 5 (𝑖 = 𝑤𝐶 = 𝑤 / 𝑖𝐶)
66 nfcv 2902 . . . . 5 𝑤𝐶
67 nfcsb1v 3932 . . . . 5 𝑖𝑤 / 𝑖𝐶
6865, 66, 67cbvsum 15727 . . . 4 Σ𝑖𝑍 𝐶 = Σ𝑤𝑍 𝑤 / 𝑖𝐶
69 csbeq1 3910 . . . . 5 (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → 𝑤 / 𝑖𝐶 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶)
70 xpfi 9355 . . . . . 6 ((𝑋 ∈ Fin ∧ 𝑌 ∈ Fin) → (𝑋 × 𝑌) ∈ Fin)
717, 14, 70syl2anc 584 . . . . 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 27251 . . . . 5 (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto𝑍)
75 fvres 6925 . . . . . 6 (𝑧 ∈ (𝑋 × 𝑌) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
7675adantl 481 . . . . 5 ((𝜑𝑧 ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
7763ralrimivva 3199 . . . . . . . . 9 (𝜑 → ∀𝑗𝑋𝑘𝑌 𝐷 ∈ ℂ)
7859eleq1d 2823 . . . . . . . . . 10 (𝑧 = ⟨𝑗, 𝑘⟩ → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ 𝐷 ∈ ℂ))
7978ralxp 5854 . . . . . . . . 9 (∀𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ∀𝑗𝑋𝑘𝑌 𝐷 ∈ ℂ)
8077, 79sylibr 234 . . . . . . . 8 (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 ∈ ℂ)
81 fveq2 6906 . . . . . . . . . . . 12 (𝑧 = 𝑤 → ( · ‘𝑧) = ( · ‘𝑤))
8281csbeq1d 3911 . . . . . . . . . . 11 (𝑧 = 𝑤( · ‘𝑧) / 𝑖𝐶 = ( · ‘𝑤) / 𝑖𝐶)
8382eleq1d 2823 . . . . . . . . . 10 (𝑧 = 𝑤 → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ( · ‘𝑤) / 𝑖𝐶 ∈ ℂ))
8483cbvralvw 3234 . . . . . . . . 9 (∀𝑧 ∈ (𝑋 × 𝑌)( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ∀𝑤 ∈ (𝑋 × 𝑌)( · ‘𝑤) / 𝑖𝐶 ∈ ℂ)
85 id 22 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌))
8682eqcoms 2742 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧( · ‘𝑧) / 𝑖𝐶 = ( · ‘𝑤) / 𝑖𝐶)
8786adantl 481 . . . . . . . . . . . . . 14 ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → ( · ‘𝑧) / 𝑖𝐶 = ( · ‘𝑤) / 𝑖𝐶)
8887eleq1d 2823 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ( · ‘𝑤) / 𝑖𝐶 ∈ ℂ))
8950eleq1d 2823 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋 × 𝑌) → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
9089adantr 480 . . . . . . . . . . . . 13 ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (( · ‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
9188, 90bitr3d 281 . . . . . . . . . . . 12 ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (( · ‘𝑤) / 𝑖𝐶 ∈ ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
9285, 91rspcdv 3613 . . . . . . . . . . 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 3142 . . . . . . . . 9 (∀𝑤 ∈ (𝑋 × 𝑌)( · ‘𝑤) / 𝑖𝐶 ∈ ℂ → ∀𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ)
9584, 94sylbi 217 . . . . . . . 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 11247 . . . . . . . . . 10 (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
98 ffn 6736 . . . . . . . . . 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 5703 . . . . . . . . . 10 ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
10136, 39, 100mp2an 692 . . . . . . . . 9 (𝑋 × 𝑌) ⊆ (ℂ × ℂ)
10269eleq1d 2823 . . . . . . . . . 10 (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → (𝑤 / 𝑖𝐶 ∈ ℂ ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ))
103102ralima 7256 . . . . . . . . 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 692 . . . . . . . 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 5701 . . . . . . . . . 10 ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))
106 f1ofo 6855 . . . . . . . . . . 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 6823 . . . . . . . . . . 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 2786 . . . . . . . . 9 (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = 𝑍)
110109raleqdv 3323 . . . . . . . 8 (𝜑 → (∀𝑤 ∈ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌))𝑤 / 𝑖𝐶 ∈ ℂ ↔ ∀𝑤𝑍 𝑤 / 𝑖𝐶 ∈ ℂ))
111104, 110bitr3id 285 . . . . . . 7 (𝜑 → (∀𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶 ∈ ℂ ↔ ∀𝑤𝑍 𝑤 / 𝑖𝐶 ∈ ℂ))
11296, 111mpbid 232 . . . . . 6 (𝜑 → ∀𝑤𝑍 𝑤 / 𝑖𝐶 ∈ ℂ)
113112r19.21bi 3248 . . . . 5 ((𝜑𝑤𝑍) → 𝑤 / 𝑖𝐶 ∈ ℂ)
11469, 71, 74, 76, 113fsumf1o 15755 . . . 4 (𝜑 → Σ𝑤𝑍 𝑤 / 𝑖𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶)
11568, 114eqtrid 2786 . . 3 (𝜑 → Σ𝑖𝑍 𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖𝐶)
11651, 64, 1153eqtr4a 2800 . 2 (𝜑 → Σ𝑗𝑋 Σ𝑘𝑌 𝐷 = Σ𝑖𝑍 𝐶)
11718, 26, 1163eqtrd 2778 1 (𝜑 → (Σ𝑗𝑋 𝐴 · Σ𝑘𝑌 𝐵) = Σ𝑖𝑍 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wral 3058  {crab 3432  csb 3907  wss 3962  cop 4636   class class class wbr 5147   × cxp 5686  ran crn 5689  cres 5690  cima 5691   Fn wfn 6557  wf 6558  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  cmpo 7432  Fincfn 8983  cc 11150  1c1 11153   · cmul 11157  cn 12263  ...cfz 13543  Σcsu 15718  cdvds 16286   gcd cgcd 16527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-dvds 16287  df-gcd 16528
This theorem is referenced by:  sgmmul  27259  dchrisum0fmul  27564
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