Theorem fsumdvdsmul 27261

Description: Product of two divisor sums. (This is also the main part of the proof that "Σ𝑘 ∥ 𝑁𝐹(𝑘) is a multiplicative function if 𝐹 is".) (Contributed by Mario Carneiro, 2-Jul-2015.) Avoid ax-mulf 11155. (Revised by GG, 18-Apr-2025.)

Hypotheses

Ref	Expression
mpodvdsmulf1o.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
mpodvdsmulf1o.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
mpodvdsmulf1o.3	⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)
mpodvdsmulf1o.x	⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
mpodvdsmulf1o.y	⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
mpodvdsmulf1o.z	⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}
fsumdvdsmul.4	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ)
fsumdvdsmul.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
fsumdvdsmul.6	⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) = 𝐷)
fsumdvdsmul.7	⊢ (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)

Assertion

Ref	Expression
fsumdvdsmul	⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 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.

Step	Hyp	Ref	Expression
1		fzfid 13988	. . . 4 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
2		mpodvdsmulf1o.x	. . . . 5 ⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
3		mpodvdsmulf1o.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		dvdsssfz1 16354	. . . . . 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 3976	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ (1...𝑀))
7	1, 6	ssfid 9215	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
8		fzfid 13988	. . . . 5 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
9		mpodvdsmulf1o.y	. . . . . 6 ⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
10		mpodvdsmulf1o.2	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11		dvdsssfz1 16354	. . . . . . 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 3976	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ (1...𝑁))
14	8, 13	ssfid 9215	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Fin)
15		fsumdvdsmul.5	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
16	14, 15	fsumcl 15762	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑌 𝐵 ∈ ℂ)
17		fsumdvdsmul.4	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ)
18	7, 16, 17	fsummulc1 15814	. 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 725	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
21	19, 17, 20	fsummulc2 15813	. . . 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 15731	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → Σ𝑘 ∈ 𝑌 (𝐴 · 𝐵) = Σ𝑘 ∈ 𝑌 𝐷)
25	21, 24	eqtrd 2799	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑘 ∈ 𝑌 𝐷)
26	25	sumeq2dv 15731	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷)
27		elxpi 5671	. . . . . . 7 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)))
28		fveq2 6869	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ = 𝑧 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
29	28	eqcoms 2772	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
30		fveq2 6869	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ = 𝑧 → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
31	30	eqcoms 2772	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
32	29, 31	eqeq12d 2780	. . . . . . . . . 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 231	. . . . . . . . 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 4037	. . . . . . . . . . . 12 ⊢ 𝑋 ⊆ ℕ
35		nnsscn 12217	. . . . . . . . . . . 12 ⊢ ℕ ⊆ ℂ
36	34, 35	sstri 3947	. . . . . . . . . . 11 ⊢ 𝑋 ⊆ ℂ
37	36	sseli 3934	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑋 → 𝑢 ∈ ℂ)
38	9	ssrab3 4037	. . . . . . . . . . . 12 ⊢ 𝑌 ⊆ ℕ
39	38, 35	sstri 3947	. . . . . . . . . . 11 ⊢ 𝑌 ⊆ ℂ
40	39	sseli 3934	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝑌 → 𝑣 ∈ ℂ)
41		ovmpot 7559	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = (𝑢 · 𝑣))
42		df-ov 7401	. . . . . . . . . . 11 ⊢ (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩)
43		df-ov 7401	. . . . . . . . . . 11 ⊢ (𝑢 · 𝑣) = ( · ‘⟨𝑢, 𝑣⟩)
44	41, 42, 43	3eqtr3g 2822	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
45	37, 40, 44	syl2an 605	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
46	33, 45	impel 513	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
47	46	exlimivv 1954	. . . . . . 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 2770	. . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
50	49	csbeq1d 3858	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
51	50	sumeq2i 15727	. . 3 ⊢ Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶
52		fveq2 6869	. . . . . . 7 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = ( · ‘⟨𝑗, 𝑘⟩))
53		df-ov 7401	. . . . . . 7 ⊢ (𝑗 · 𝑘) = ( · ‘⟨𝑗, 𝑘⟩)
54	52, 53	eqtr4di 2817	. . . . . 6 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = (𝑗 · 𝑘))
55	54	csbeq1d 3858	. . . . 5 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
56		ovex 7431	. . . . . 6 ⊢ (𝑗 · 𝑘) ∈ V
57		fsumdvdsmul.7	. . . . . 6 ⊢ (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)
58	56, 57	csbie 3889	. . . . 5 ⊢ ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 = 𝐷
59	55, 58	eqtrdi 2815	. . . 4 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = 𝐷)
60	17	adantrr 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐴 ∈ ℂ)
61	15	adantrl 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐵 ∈ ℂ)
62	60, 61	mulcld 11204	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) ∈ ℂ)
63	22, 62	eqeltrrd 2865	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐷 ∈ ℂ)
64	59, 7, 14, 63	fsumxp 15801	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶)
65		csbeq1a 3868	. . . . 5 ⊢ (𝑖 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑖⦌𝐶)
66		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑤𝐶
67		nfcsb1v 3878	. . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑖⦌𝐶
68	65, 66, 67	cbvsum 15724	. . . 4 ⊢ Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶
69		csbeq1 3857	. . . . 5 ⊢ (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → ⦋𝑤 / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
70		xpfi 9266	. . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑌 ∈ Fin) → (𝑋 × 𝑌) ∈ Fin)
71	7, 14, 70	syl2anc 593	. . . . 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 27260	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)
75		fvres 6888	. . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
76	75	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
77	63	ralrimivva 3207	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 𝐷 ∈ ℂ)
78	59	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ 𝐷 ∈ ℂ))
79	78	ralxp 5815	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 𝐷 ∈ ℂ)
80	77, 79	sylibr 236	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)
81		fveq2 6869	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ( · ‘𝑧) = ( · ‘𝑤))
82	81	csbeq1d 3858	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
83	82	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ))
84	83	cbvralvw 3242	. . . . . . . . 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 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
87	86	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
88	87	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ))
89	50	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
90	89	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
91	88, 90	bitr3d 283	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
92	85, 91	rspcdv 3575	. . . . . . . . . . 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 3155	. . . . . . . . 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 219	. . . . . . . 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 11170	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
98		ffn 6693	. . . . . . . . . 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 5664	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
101	36, 39, 100	mp2an 702	. . . . . . . . 9 ⊢ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)
102	69	eleq1d 2849	. . . . . . . . . 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 7223	. . . . . . . . 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 702	. . . . . . . 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 5662	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))
106		f1ofo 6816	. . . . . . . . . . 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 6783	. . . . . . . . . . 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 2811	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = 𝑍)
110	109	raleqdv 3322	. . . . . . . 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 287	. . . . . . 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 234	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)
113	112	r19.21bi 3256	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)
114	69, 71, 74, 76, 113	fsumf1o 15752	. . . 4 ⊢ (𝜑 → Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
115	68, 114	eqtrid 2811	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
116	51, 64, 115	3eqtr4a 2825	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑖 ∈ 𝑍 𝐶)
117	18, 26, 116	3eqtrd 2803	1 ⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∀wral 3078  {crab 3416  ⦋csb 3854   ⊆ wss 3906  ⟨cop 4590   class class class wbr 5102   × cxp 5647  ran crn 5650   ↾ cres 5651   “ cima 5652   Fn wfn 6518  ⟶wf 6519  –onto→wfo 6521  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  Fincfn 8929  ℂcc 11073  1c1 11076   · cmul 11080  ℕcn 12212  ...cfz 13514  Σcsu 15715   ∥ cdvds 16288   gcd cgcd 16530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-z 12571  df-uz 12842  df-rp 12996  df-fz 13515  df-fzo 13662  df-fl 13804  df-mod 13882  df-seq 14017  df-exp 14077  df-hash 14346  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-clim 15517  df-sum 15716  df-dvds 16289  df-gcd 16531

This theorem is referenced by:  sgmmul  27267  dchrisum0fmul  27572