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Theorem sgmmul 25125

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 1234	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → 𝑀 ∈ ℕ)
2		simpr2 1236	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → 𝑁 ∈ ℕ)
3		simpr3 1238	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑀 gcd 𝑁) = 1)
4		eqid 2760	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
5		eqid 2760	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
6		eqid 2760	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}
7		ssrab2 3828	. . . . . 6 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} ⊆ ℕ
8		simpr 479	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀})
9	7, 8	sseldi 3742	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝑗 ∈ ℕ)
10	9	nncnd 11228	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝑗 ∈ ℂ)
11		simpll 807	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝐴 ∈ ℂ)
12	10, 11	cxpcld 24653	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → (𝑗↑_𝑐𝐴) ∈ ℂ)
13		ssrab2 3828	. . . . . 6 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ ℕ
14		simpr 479	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁})
15	13, 14	sseldi 3742	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ ℕ)
16	15	nncnd 11228	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ ℂ)
17		simpll 807	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝐴 ∈ ℂ)
18	16, 17	cxpcld 24653	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑘↑_𝑐𝐴) ∈ ℂ)
19	9	adantrr 755	. . . . . 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 11227	. . . . 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 11543	. . . . . 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 11546	. . . . 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 754	. . . . . 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 11227	. . . . 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 11543	. . . . . 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 11546	. . . . 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 807	. . . . 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 24673	. . . 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 2766	. . 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 6820	. . 3 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝑖↑_𝑐𝐴) = ((𝑗 · 𝑘)↑_𝑐𝐴))
31	1, 2, 3, 4, 5, 6, 12, 18, 29, 30	fsumdvdsmul 25120	. 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 25067	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ) → (𝐴 σ 𝑀) = Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} (𝑗↑_𝑐𝐴))
33	1, 32	syldan 488	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ 𝑀) = Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} (𝑗↑_𝑐𝐴))
34		sgmval 25067	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 σ 𝑁) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑘↑_𝑐𝐴))
35	2, 34	syldan 488	. . 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 6831	. 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 11260	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑀 · 𝑁) ∈ ℕ)
38		sgmval 25067	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 · 𝑁) ∈ ℕ) → (𝐴 σ (𝑀 · 𝑁)) = Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} (𝑖↑_𝑐𝐴))
39	37, 38	syldan 488	. 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 2805	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 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 {crab 3054 class class class wbr 4804 (class class class)co 6813 ℂcc 10126 1c1 10129 · cmul 10133 ℕcn 11212 Σcsu 14615 ∥ cdvds 15182 gcd cgcd 15418 ↑_𝑐ccxp 24501 σ csgm 25021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ioc 12373 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-mod 12863 df-seq 12996 df-exp 13055 df-fac 13255 df-bc 13284 df-hash 13312 df-shft 14006 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-limsup 14401 df-clim 14418 df-rlim 14419 df-sum 14616 df-ef 14997 df-sin 14999 df-cos 15000 df-pi 15002 df-dvds 15183 df-gcd 15419 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-pt 16307 df-prds 16310 df-xrs 16364 df-qtop 16369 df-imas 16370 df-xps 16372 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-fbas 19945 df-fg 19946 df-cnfld 19949 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 df-nei 21104 df-lp 21142 df-perf 21143 df-cn 21233 df-cnp 21234 df-haus 21321 df-tx 21567 df-hmeo 21760 df-fil 21851 df-fm 21943 df-flim 21944 df-flf 21945 df-xms 22326 df-ms 22327 df-tms 22328 df-cncf 22882 df-limc 23829 df-dv 23830 df-log 24502 df-cxp 24503 df-sgm 25027

This theorem is referenced by: perfect1 25152 perfectlem1 25153 perfectlem2 25154 perfectALTVlem1 42140 perfectALTVlem2 42141

W3C validator