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Theorem sgmmul 26254

Description: The divisor function for fixed parameter 𝐴 is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)

**Assertion**
Ref	Expression
sgmmul	⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ (𝑀 · 𝑁)) = ((𝐴 σ 𝑀) · (𝐴 σ 𝑁)))

Proof of Theorem sgmmul Dummy variables 𝑖 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr1 1192	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → 𝑀 ∈ ℕ)
2		simpr2 1193	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → 𝑁 ∈ ℕ)
3		simpr3 1194	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑀 gcd 𝑁) = 1)
4		eqid 2738	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
5		eqid 2738	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
6		eqid 2738	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}
7		ssrab2 4009	. . . . . 6 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ⊆ ℕ
8		simpr 484	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀})
9	7, 8	sselid 3915	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝑗 ∈ ℕ)
10	9	nncnd 11919	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝑗 ∈ ℂ)
11		simpll 763	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝐴 ∈ ℂ)
12	10, 11	cxpcld 25768	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → (𝑗↑_𝑐𝐴) ∈ ℂ)
13		ssrab2 4009	. . . . . 6 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ ℕ
14		simpr 484	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
15	13, 14	sselid 3915	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ ℕ)
16	15	nncnd 11919	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ ℂ)
17		simpll 763	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝐴 ∈ ℂ)
18	16, 17	cxpcld 25768	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑘↑_𝑐𝐴) ∈ ℂ)
19	9	adantrr 713	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 𝑗 ∈ ℕ)
20	19	nnred 11918	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 𝑗 ∈ ℝ)
21	19	nnnn0d 12223	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 𝑗 ∈ ℕ₀)
22	21	nn0ge0d 12226	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 0 ≤ 𝑗)
23	15	adantrl 712	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 𝑘 ∈ ℕ)
24	23	nnred 11918	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 𝑘 ∈ ℝ)
25	23	nnnn0d 12223	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 𝑘 ∈ ℕ₀)
26	25	nn0ge0d 12226	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 0 ≤ 𝑘)
27		simpll 763	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → 𝐴 ∈ ℂ)
28	20, 22, 24, 26, 27	mulcxpd 25788	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → ((𝑗 · 𝑘)↑_𝑐𝐴) = ((𝑗↑_𝑐𝐴) · (𝑘↑_𝑐𝐴)))
29	28	eqcomd 2744	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})) → ((𝑗↑_𝑐𝐴) · (𝑘↑_𝑐𝐴)) = ((𝑗 · 𝑘)↑_𝑐𝐴))
30		oveq1 7262	. . 3 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝑖↑_𝑐𝐴) = ((𝑗 · 𝑘)↑_𝑐𝐴))
31	1, 2, 3, 4, 5, 6, 12, 18, 29, 30	fsumdvdsmul 26249	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} (𝑗↑_𝑐𝐴) · Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑘↑_𝑐𝐴)) = Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} (𝑖↑_𝑐𝐴))
32		sgmval 26196	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ) → (𝐴 σ 𝑀) = Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} (𝑗↑_𝑐𝐴))
33	1, 32	syldan 590	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ 𝑀) = Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} (𝑗↑_𝑐𝐴))
34		sgmval 26196	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 σ 𝑁) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑘↑_𝑐𝐴))
35	2, 34	syldan 590	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ 𝑁) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑘↑_𝑐𝐴))
36	33, 35	oveq12d 7273	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → ((𝐴 σ 𝑀) · (𝐴 σ 𝑁)) = (Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} (𝑗↑_𝑐𝐴) · Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑘↑_𝑐𝐴)))
37	1, 2	nnmulcld 11956	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑀 · 𝑁) ∈ ℕ)
38		sgmval 26196	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 · 𝑁) ∈ ℕ) → (𝐴 σ (𝑀 · 𝑁)) = Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} (𝑖↑_𝑐𝐴))
39	37, 38	syldan 590	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ (𝑀 · 𝑁)) = Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} (𝑖↑_𝑐𝐴))
40	31, 36, 39	3eqtr4rd 2789	1 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ (𝑀 · 𝑁)) = ((𝐴 σ 𝑀) · (𝐴 σ 𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 {crab 3067 class class class wbr 5070 (class class class)co 7255 ℂcc 10800 1c1 10803 · cmul 10807 ℕcn 11903 Σcsu 15325 ∥ cdvds 15891 gcd cgcd 16129 ↑_𝑐ccxp 25616 σ csgm 26150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-dvds 15892 df-gcd 16130 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618 df-sgm 26156

This theorem is referenced by: perfect1 26281 perfectlem1 26282 perfectlem2 26283 perfectALTVlem1 45061 perfectALTVlem2 45062

W3C validator