Theorem fsumdvdsmul 15650

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 8110. (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 12747	. . . . 5 ⊢ (𝑀 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∈ Fin)
4	2, 3	syl 14	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∈ Fin)
5	1, 4	eqeltrid 2316	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
6		mpodvdsmulf1o.y	. . . . 5 ⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
7		mpodvdsmulf1o.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8		dvdsfi 12747	. . . . . 6 ⊢ (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin)
9	7, 8	syl 14	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin)
10	6, 9	eqeltrid 2316	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Fin)
11		fsumdvdsmul.5	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
12	10, 11	fsumcl 11897	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑌 𝐵 ∈ ℂ)
13		fsumdvdsmul.4	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ)
14	5, 12, 13	fsummulc1 11946	. 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 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)
17	15, 13, 16	fsummulc2 11945	. . . 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 400	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋) ∧ 𝑘 ∈ 𝑌) → (𝐴 · 𝐵) = 𝐷)
20	19	sumeq2dv 11865	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → Σ𝑘 ∈ 𝑌 (𝐴 · 𝐵) = Σ𝑘 ∈ 𝑌 𝐷)
21	17, 20	eqtrd 2262	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑘 ∈ 𝑌 𝐷)
22	21	sumeq2dv 11865	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 (𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷)
23		elxpi 4732	. . . . . . 7 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌)))
24		fveq2 5623	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ = 𝑧 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
25	24	eqcoms 2232	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
26		fveq2 5623	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ = 𝑧 → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
27	26	eqcoms 2232	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ( · ‘⟨𝑢, 𝑣⟩) = ( · ‘𝑧))
28	25, 27	eqeq12d 2244	. . . . . . . . . 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 3310	. . . . . . . . . . . 12 ⊢ 𝑋 ⊆ ℕ
31		nnsscn 9103	. . . . . . . . . . . 12 ⊢ ℕ ⊆ ℂ
32	30, 31	sstri 3233	. . . . . . . . . . 11 ⊢ 𝑋 ⊆ ℂ
33	32	sseli 3220	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑋 → 𝑢 ∈ ℂ)
34	6	ssrab3 3310	. . . . . . . . . . . 12 ⊢ 𝑌 ⊆ ℕ
35	34, 31	sstri 3233	. . . . . . . . . . 11 ⊢ 𝑌 ⊆ ℂ
36	35	sseli 3220	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝑌 → 𝑣 ∈ ℂ)
37		mulcl 8114	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
38		oveq1 6001	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
39		oveq2 6002	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
40		eqid 2229	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))
41	38, 39, 40	ovmpog 6130	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ ∧ (𝑢 · 𝑣) ∈ ℂ) → (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = (𝑢 · 𝑣))
42	37, 41	mpd3an3 1372	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = (𝑢 · 𝑣))
43		df-ov 5997	. . . . . . . . . . 11 ⊢ (𝑢(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝑣) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘⟨𝑢, 𝑣⟩)
44		df-ov 5997	. . . . . . . . . . 11 ⊢ (𝑢 · 𝑣) = ( · ‘⟨𝑢, 𝑣⟩)
45	42, 43, 44	3eqtr3g 2285	. . . . . . . . . 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 1943	. . . . . . 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 2235	. . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ( · ‘𝑧) = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧))
51	50	csbeq1d 3131	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
52	51	sumeq2i 11861	. . 3 ⊢ Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶
53		simprl 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝑗 ∈ 𝑋)
54	30, 53	sselid 3222	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝑗 ∈ ℕ)
55		simprr 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝑘 ∈ 𝑌)
56	34, 55	sselid 3222	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝑘 ∈ ℕ)
57	54, 56	nnmulcld 9147	. . . . . . . 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 3171	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 = 𝐷)
61		anass 401	. . . . . . . 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 2231	. . . . . . 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 11865	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
66	65	sumeq2dv 11865	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
67		fveq2 5623	. . . . . . 7 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = ( · ‘⟨𝑗, 𝑘⟩))
68		df-ov 5997	. . . . . . 7 ⊢ (𝑗 · 𝑘) = ( · ‘⟨𝑗, 𝑘⟩)
69	67, 68	eqtr4di 2280	. . . . . 6 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ( · ‘𝑧) = (𝑗 · 𝑘))
70	69	csbeq1d 3131	. . . . 5 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶)
71	13	adantrr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐴 ∈ ℂ)
72	11	adantrl 478	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐵 ∈ ℂ)
73	71, 72	mulcld 8155	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) ∈ ℂ)
74	18, 73	eqeltrrd 2307	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → 𝐷 ∈ ℂ)
75	60, 74	eqeltrd 2306	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 ∈ ℂ)
76	70, 5, 10, 75	fsumxp 11933	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶)
77	66, 76	eqtrd 2262	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋( · ‘𝑧) / 𝑖⦌𝐶)
78		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑤𝐶
79		nfcsb1v 3157	. . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑖⦌𝐶
80		csbeq1a 3133	. . . . 5 ⊢ (𝑖 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑖⦌𝐶)
81	78, 79, 80	cbvsumi 11859	. . . 4 ⊢ Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶
82		csbeq1 3127	. . . . 5 ⊢ (𝑤 = ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) → ⦋𝑤 / 𝑖⦌𝐶 = ⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
83		xpfi 7082	. . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑌 ∈ Fin) → (𝑋 × 𝑌) ∈ Fin)
84	5, 10, 83	syl2anc 411	. . . . 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 15649	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)
88		fvres 5647	. . . . . 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 2612	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝑋 ∀𝑘 ∈ 𝑌 ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 ∈ ℂ)
91	70	eleq1d 2298	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋(𝑗 · 𝑘) / 𝑖⦌𝐶 ∈ ℂ))
92	91	ralxp 4862	. . . . . . . . 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 5623	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ( · ‘𝑧) = ( · ‘𝑤))
95	94	csbeq1d 3131	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ⦋( · ‘𝑧) / 𝑖⦌𝐶 = ⦋( · ‘𝑤) / 𝑖⦌𝐶)
96	95	eleq1d 2298	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ))
97	96	cbvralvw 2769	. . . . . . . . 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 2232	. . . . . . . . . . . . . . 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 2298	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑤 = 𝑧) → (⦋( · ‘𝑧) / 𝑖⦌𝐶 ∈ ℂ ↔ ⦋( · ‘𝑤) / 𝑖⦌𝐶 ∈ ℂ))
102	51	eleq1d 2298	. . . . . . . . . . . . . 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 2910	. . . . . . . . . . 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 2602	. . . . . . . . 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 8124	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
111		ffn 5469	. . . . . . . . . 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 4823	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
114	32, 35, 113	mp2an 426	. . . . . . . . 9 ⊢ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)
115	82	eleq1d 2298	. . . . . . . . . 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 5872	. . . . . . . . 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 426	. . . . . . . 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 4729	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = ran ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌))
119		f1ofo 5575	. . . . . . . . . . 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 5547	. . . . . . . . . . 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 2274	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) “ (𝑋 × 𝑌)) = 𝑍)
123	122	raleqdv 2734	. . . . . . . 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 2618	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ⦋𝑤 / 𝑖⦌𝐶 ∈ ℂ)
127	82, 84, 87, 89, 126	fsumf1o 11887	. . . 4 ⊢ (𝜑 → Σ𝑤 ∈ 𝑍 ⦋𝑤 / 𝑖⦌𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
128	81, 127	eqtrid 2274	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ 𝑍 𝐶 = Σ𝑧 ∈ (𝑋 × 𝑌)⦋((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))‘𝑧) / 𝑖⦌𝐶)
129	52, 77, 128	3eqtr4a 2288	. 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑋 Σ𝑘 ∈ 𝑌 𝐷 = Σ𝑖 ∈ 𝑍 𝐶)
130	14, 22, 129	3eqtrd 2266	1 ⊢ (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  {crab 2512  ⦋csb 3124   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4082   × cxp 4714  ran crn 4717   ↾ cres 4718   “ cima 4719   Fn wfn 5309  ⟶wf 5310  –onto→wfo 5312  –1-1-onto→wf1o 5313  ‘cfv 5314  (class class class)co 5994   ∈ cmpo 5996  Fincfn 6877  ℂcc 7985  1c1 7988   · cmul 7992  ℕcn 9098  Σcsu 11850   ∥ cdvds 12284   gcd cgcd 12460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-disj 4059  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-frec 6527  df-1o 6552  df-oadd 6556  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-sup 7139  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-fl 10477  df-mod 10532  df-seqfrec 10657  df-exp 10748  df-ihash 10985  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-clim 11776  df-sumdc 11851  df-dvds 12285  df-gcd 12461

This theorem is referenced by:  sgmmul  15655