Theorem fsumdvdsmul 27165

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 11110. (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 13900	. . . 4 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
2		mpodvdsmulf1o.x	. . . . 5 ⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
3		mpodvdsmulf1o.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		dvdsssfz1 16249	. . . . . 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 3973	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ (1...𝑀))
7	1, 6	ssfid 9173	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
8		fzfid 13900	. . . . 5 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
9		mpodvdsmulf1o.y	. . . . . 6 ⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
10		mpodvdsmulf1o.2	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
11		dvdsssfz1 16249	. . . . . . 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 3973	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ (1...𝑁))
14	8, 13	ssfid 9173	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Fin)
15		fsumdvdsmul.5	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
16	14, 15	fsumcl 15660	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑌 𝐵 ∈ ℂ)
17		fsumdvdsmul.4	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ)
18	7, 16, 17	fsummulc1 15712	. 2 ⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵))
19	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑌 ∈ Fin)
20	15	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
21	19, 17, 20	fsummulc2 15711	. . . 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 467	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → (𝐴 · 𝐵) = 𝐷)
24	23	sumeq2dv 15629	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → Σ𝑘 ∈ 𝑌 (𝐴 · 𝐵) = Σ𝑘 ∈ 𝑌 𝐷)
25	21, 24	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑘 ∈ 𝑌 𝐷)
26	25	sumeq2dv 15629	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷)
27		elxpi 5647	. . . . . . 7 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)))
28		fveq2 6835	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ = 𝑧 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
29	28	eqcoms 2745	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
30		fveq2 6835	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ = 𝑧 → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
31	30	eqcoms 2745	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
32	29, 31	eqeq12d 2753	. . . . . . . . . 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 229	. . . . . . . . 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 4035	. . . . . . . . . . . 12 ⊢ 𝑋 ⊆ ℕ
35		nnsscn 12154	. . . . . . . . . . . 12 ⊢ ℕ ⊆ ℂ
36	34, 35	sstri 3944	. . . . . . . . . . 11 ⊢ 𝑋 ⊆ ℂ
37	36	sseli 3930	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑋 → 𝑢 ∈ ℂ)
38	9	ssrab3 4035	. . . . . . . . . . . 12 ⊢ 𝑌 ⊆ ℕ
39	38, 35	sstri 3944	. . . . . . . . . . 11 ⊢ 𝑌 ⊆ ℂ
40	39	sseli 3930	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝑌 → 𝑣 ∈ ℂ)
41		ovmpot 7521	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = (𝑢 · 𝑣))
42		df-ov 7363	. . . . . . . . . . 11 ⊢ (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩)
43		df-ov 7363	. . . . . . . . . . 11 ⊢ (𝑢 · 𝑣) = ( · ‘⟨𝑢, 𝑣⟩)
44	41, 42, 43	3eqtr3g 2795	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
45	37, 40, 44	syl2an 597	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
46	33, 45	impel 505	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
47	46	exlimivv 1934	. . . . . . 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 2743	. . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
50	49	csbeq1d 3854	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
51	50	sumeq2i 15625	. . 3 ⊢ Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶
52		fveq2 6835	. . . . . . 7 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = ( · ‘⟨𝑗, 𝑘⟩))
53		df-ov 7363	. . . . . . 7 ⊢ (𝑗 · 𝑘) = ( · ‘⟨𝑗, 𝑘⟩)
54	52, 53	eqtr4di 2790	. . . . . 6 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = (𝑗 · 𝑘))
55	54	csbeq1d 3854	. . . . 5 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
56		ovex 7393	. . . . . 6 ⊢ (𝑗 · 𝑘) ∈ V
57		fsumdvdsmul.7	. . . . . 6 ⊢ (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)
58	56, 57	csbie 3885	. . . . 5 ⊢ ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 = 𝐷
59	55, 58	eqtrdi 2788	. . . 4 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = 𝐷)
60	17	adantrr 718	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐴 ∈ ℂ)
61	15	adantrl 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐵 ∈ ℂ)
62	60, 61	mulcld 11156	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) ∈ ℂ)
63	22, 62	eqeltrrd 2838	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐷 ∈ ℂ)
64	59, 7, 14, 63	fsumxp 15699	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶)
65		csbeq1a 3864	. . . . 5 ⊢ (𝑖 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑖⦌𝐶)
66		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑤𝐶
67		nfcsb1v 3874	. . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑖⦌𝐶
68	65, 66, 67	cbvsum 15622	. . . 4 ⊢ Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶
69		csbeq1 3853	. . . . 5 ⊢ (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → ⦋𝑤 / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
70		xpfi 9224	. . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑌 ∈ Fin) → (𝑋 × 𝑌) ∈ Fin)
71	7, 14, 70	syl2anc 585	. . . . 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 27164	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)
75		fvres 6854	. . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
76	75	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
77	63	ralrimivva 3180	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 𝐷 ∈ ℂ)
78	59	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ 𝐷 ∈ ℂ))
79	78	ralxp 5791	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 𝐷 ∈ ℂ)
80	77, 79	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)
81		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ( · ‘𝑧) = ( · ‘𝑤))
82	81	csbeq1d 3854	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
83	82	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ))
84	83	cbvralvw 3215	. . . . . . . . 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 2745	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
87	86	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
88	87	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ))
89	50	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
90	89	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
91	88, 90	bitr3d 281	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
92	85, 91	rspcdv 3569	. . . . . . . . . . 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 3128	. . . . . . . . 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 217	. . . . . . . 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 11125	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
98		ffn 6663	. . . . . . . . . 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 5640	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
101	36, 39, 100	mp2an 693	. . . . . . . . 9 ⊢ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)
102	69	eleq1d 2822	. . . . . . . . . 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 7185	. . . . . . . . 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 693	. . . . . . . 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 5638	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))
106		f1ofo 6782	. . . . . . . . . . 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 6750	. . . . . . . . . . 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 2784	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = 𝑍)
110	109	raleqdv 3297	. . . . . . . 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 285	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)
113	112	r19.21bi 3229	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)
114	69, 71, 74, 76, 113	fsumf1o 15650	. . . 4 ⊢ (𝜑 → Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
115	68, 114	eqtrid 2784	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
116	51, 64, 115	3eqtr4a 2798	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑖 ∈ 𝑍 𝐶)
117	18, 26, 116	3eqtrd 2776	1 ⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  {crab 3400  ⦋csb 3850   ⊆ wss 3902  ⟨cop 4587   class class class wbr 5099   × cxp 5623  ran crn 5626   ↾ cres 5627   “ cima 5628   Fn wfn 6488  ⟶wf 6489  –onto→wfo 6491  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362  Fincfn 8887  ℂcc 11028  1c1 11031   · cmul 11035  ℕcn 12149  ...cfz 13427  Σcsu 15613   ∥ cdvds 16183   gcd cgcd 16425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-n0 12406  df-z 12493  df-uz 12756  df-rp 12910  df-fz 13428  df-fzo 13575  df-fl 13716  df-mod 13794  df-seq 13929  df-exp 13989  df-hash 14258  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-clim 15415  df-sum 15614  df-dvds 16184  df-gcd 16426

This theorem is referenced by:  sgmmul  27172  dchrisum0fmul  27477