Theorem fsumdvdsmul 16297

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 8309. (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		mpodvdsmulf1o.x	. . . 4 ⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
2		mpodvdsmulf1o.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		dvdsfi 13047	. . . . 5 ⊢ (𝑀 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∈ Fin)
4	2, 3	syl 14	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∈ Fin)
5	1, 4	eqeltrid 2325	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
6		mpodvdsmulf1o.y	. . . . 5 ⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
7		mpodvdsmulf1o.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8		dvdsfi 13047	. . . . . 6 ⊢ (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin)
9	7, 8	syl 14	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin)
10	6, 9	eqeltrid 2325	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Fin)
11		fsumdvdsmul.5	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
12	10, 11	fsumcl 12193	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑌 𝐵 ∈ ℂ)
13		fsumdvdsmul.4	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ)
14	5, 12, 13	fsummulc1 12242	. 2 ⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵))
15	10	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝑌 ∈ Fin)
16	11	adantlr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
17	15, 13, 16	fsummulc2 12241	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑘 ∈ 𝑌 (𝐴 · 𝐵))
18		fsumdvdsmul.6	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) = 𝐷)
19	18	anassrs 404	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → (𝐴 · 𝐵) = 𝐷)
20	19	sumeq2dv 12160	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → Σ𝑘 ∈ 𝑌 (𝐴 · 𝐵) = Σ𝑘 ∈ 𝑌 𝐷)
21	17, 20	eqtrd 2271	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑘 ∈ 𝑌 𝐷)
22	21	sumeq2dv 12160	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷)
23		elxpi 4792	. . . . . . 7 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)))
24		fveq2 5698	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ = 𝑧 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
25	24	eqcoms 2241	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
26		fveq2 5698	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ = 𝑧 → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
27	26	eqcoms 2241	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
28	25, 27	eqeq12d 2253	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩) ↔ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)))
29	28	biimpd 144	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧)))
30	1	ssrab3 3334	. . . . . . . . . . . 12 ⊢ 𝑋 ⊆ ℕ
31		nnsscn 9319	. . . . . . . . . . . 12 ⊢ ℕ ⊆ ℂ
32	30, 31	sstri 3257	. . . . . . . . . . 11 ⊢ 𝑋 ⊆ ℂ
33	32	sseli 3244	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑋 → 𝑢 ∈ ℂ)
34	6	ssrab3 3334	. . . . . . . . . . . 12 ⊢ 𝑌 ⊆ ℕ
35	34, 31	sstri 3257	. . . . . . . . . . 11 ⊢ 𝑌 ⊆ ℂ
36	35	sseli 3244	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝑌 → 𝑣 ∈ ℂ)
37		mulcl 8313	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
38		oveq1 6095	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
39		oveq2 6096	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
40		eqid 2238	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))
41	38, 39, 40	ovmpog 6226	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ ∧ (𝑢 · 𝑣) ∈ ℂ) → (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = (𝑢 · 𝑣))
42	37, 41	mpd3an3 1379	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = (𝑢 · 𝑣))
43		df-ov 6091	. . . . . . . . . . 11 ⊢ (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩)
44		df-ov 6091	. . . . . . . . . . 11 ⊢ (𝑢 · 𝑣) = ( · ‘⟨𝑢, 𝑣⟩)
45	42, 43, 44	3eqtr3g 2294	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
46	33, 36, 45	syl2an 289	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ( · ‘⟨𝑢, 𝑣⟩))
47	29, 46	impel 280	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
48	47	exlimivv 1952	. . . . . . 7 ⊢ (∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
49	23, 48	syl 14	. . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) = ( · ‘𝑧))
50	49	eqcomd 2244	. . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
51	50	csbeq1d 3154	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
52	51	sumeq2i 12156	. . 3 ⊢ Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶
53		simprl 535	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝑗 ∈ 𝑋)
54	30, 53	sselid 3246	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝑗 ∈ ℕ)
55		simprr 537	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝑘 ∈ 𝑌)
56	34, 55	sselid 3246	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝑘 ∈ ℕ)
57	54, 56	nnmulcld 9363	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝑗 · 𝑘) ∈ ℕ)
58		fsumdvdsmul.7	. . . . . . . . 9 ⊢ (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)
59	58	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) ∧ 𝑖 = (𝑗 · 𝑘)) → 𝐶 = 𝐷)
60	57, 59	csbied 3194	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 = 𝐷)
61		anass 405	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) ↔ (𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)))
62	61	bicomi 132	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌))
63		eqcom 2240	. . . . . . 7 ⊢ (⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 = 𝐷 ↔ 𝐷 = ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
64	60, 62, 63	3imtr3i 200	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → 𝐷 = ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
65	64	sumeq2dv 12160	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
66	65	sumeq2dv 12160	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
67		fveq2 5698	. . . . . . 7 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = ( · ‘⟨𝑗, 𝑘⟩))
68		df-ov 6091	. . . . . . 7 ⊢ (𝑗 · 𝑘) = ( · ‘⟨𝑗, 𝑘⟩)
69	67, 68	eqtr4di 2289	. . . . . 6 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = (𝑗 · 𝑘))
70	69	csbeq1d 3154	. . . . 5 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
71	13	adantrr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐴 ∈ ℂ)
72	11	adantrl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐵 ∈ ℂ)
73	71, 72	mulcld 8353	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) ∈ ℂ)
74	18, 73	eqeltrrd 2316	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐷 ∈ ℂ)
75	60, 74	eqeltrd 2315	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 ∈ ℂ)
76	70, 5, 10, 75	fsumxp 12229	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶)
77	66, 76	eqtrd 2271	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶)
78		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑤𝐶
79		nfcsb1v 3180	. . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑖⦌𝐶
80		csbeq1a 3156	. . . . 5 ⊢ (𝑖 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑖⦌𝐶)
81	78, 79, 80	cbvsumi 12154	. . . 4 ⊢ Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶
82		csbeq1 3150	. . . . 5 ⊢ (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → ⦋𝑤 / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
83		xpfi 7244	. . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑌 ∈ Fin) → (𝑋 × 𝑌) ∈ Fin)
84	5, 10, 83	syl2anc 415	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ Fin)
85		mpodvdsmulf1o.3	. . . . . 6 ⊢ (𝜑 → (𝑀 gcd 𝑁) = 1)
86		mpodvdsmulf1o.z	. . . . . 6 ⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}
87	2, 7, 85, 1, 6, 86	mpodvdsmulf1o 16296	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)
88		fvres 5722	. . . . . 6 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
89	88	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
90	75	ralrimivva 2632	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 ∈ ℂ)
91	70	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 ∈ ℂ))
92	91	ralxp 4925	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 ∈ ℂ)
93	90, 92	sylibr 134	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)
94		fveq2 5698	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ( · ‘𝑧) = ( · ‘𝑤))
95	94	csbeq1d 3154	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
96	95	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ))
97	96	cbvralvw 2790	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑤 ∈ (𝑋 × 𝑌)⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ)
98		id 19	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌))
99	95	eqcoms 2241	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
100	99	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
101	100	eleq1d 2307	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ))
102	51	eleq1d 2307	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
103	102	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
104	101, 103	bitr3d 190	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
105	98, 104	rspcdv 2932	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (∀𝑤 ∈ (𝑋 × 𝑌)⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ → ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
106	105	com12 30	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ (𝑋 × 𝑌)⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ → (𝑧 ∈ (𝑋 × 𝑌) → ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
107	106	ralrimiv 2622	. . . . . . . . 9 ⊢ (∀𝑤 ∈ (𝑋 × 𝑌)⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ → ∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)
108	97, 107	sylbi 121	. . . . . . . 8 ⊢ (∀𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ → ∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)
109	93, 108	syl 14	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ)
110		mpomulf 8323	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
111		ffn 5536	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ → (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ))
112	110, 111	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) Fn (ℂ × ℂ)
113		xpss12 4884	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
114	32, 35, 113	mp2an 430	. . . . . . . . 9 ⊢ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)
115	82	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → (⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ))
116	115	ralima 5964	. . . . . . . . 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))
117	112, 114, 116	mp2an 430	. . . . . . . 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)
118		df-ima 4789	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))
119		f1ofo 5649	. . . . . . . . . . 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)
120		forn 5621	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍 → ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) = 𝑍)
121	87, 119, 120	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)) = 𝑍)
122	118, 121	eqtrid 2283	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = 𝑍)
123	122	raleqdv 2755	. . . . . . . 8 ⊢ (𝜑 → (∀𝑤 ∈ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌))⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ))
124	117, 123	bitr3id 194	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ))
125	109, 124	mpbid 147	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)
126	125	r19.21bi 2638	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)
127	82, 84, 87, 89, 126	fsumf1o 12183	. . . 4 ⊢ (𝜑 → Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
128	81, 127	eqtrid 2283	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
129	52, 77, 128	3eqtr4a 2297	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑖 ∈ 𝑍 𝐶)
130	14, 22, 129	3eqtrd 2275	1 ⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  ⦋csb 3147   ⊆ wss 3220  ⟨cop 3712   class class class wbr 4131   × cxp 4774  ran crn 4777   ↾ cres 4778   “ cima 4779   Fn wfn 5375  ⟶wf 5376  –onto→wfo 5378  –1-1-onto→wf1o 5379  ‘cfv 5380  (class class class)co 6088   ∈ cmpo 6090  Fincfn 7027  ℂcc 8184  1c1 8187   · cmul 8191  ℕcn 9314  Σcsu 12145   ∥ cdvds 12580   gcd cgcd 12756

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-fz 10430  df-fzo 10568  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-ihash 11238  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-sumdc 12146  df-dvds 12581  df-gcd 12757

This theorem is used by:  sgmmul  16302