sgmmul - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sgmmul		Structured version Visualization version GIF version

Theorem sgmmul 27128

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 1195	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → 𝑀 ∈ ℕ)
2		simpr2 1196	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → 𝑁 ∈ ℕ)
3		simpr3 1197	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑀 gcd 𝑁) = 1)
4		eqid 2729	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}
5		eqid 2729	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}
6		eqid 2729	. . 3 ⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}
7		ssrab2 4033	. . . . . 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 3935	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝑗 ∈ ℕ)
10	9	nncnd 12162	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝑗 ∈ ℂ)
11		simpll 766	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → 𝐴 ∈ ℂ)
12	10, 11	cxpcld 26633	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}) → (𝑗↑_𝑐𝐴) ∈ ℂ)
13		ssrab2 4033	. . . . . 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 3935	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ ℕ)
16	15	nncnd 12162	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ ℂ)
17		simpll 766	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝐴 ∈ ℂ)
18	16, 17	cxpcld 26633	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑘↑_𝑐𝐴) ∈ ℂ)
19	9	adantrr 717	. . . . . 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 12161	. . . . 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 12463	. . . . . 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 12466	. . . . 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 716	. . . . . 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 12161	. . . . 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 12463	. . . . . 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 12466	. . . . 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 766	. . . . 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 26653	. . . 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 2735	. . 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 7360	. . 3 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝑖↑_𝑐𝐴) = ((𝑗 · 𝑘)↑_𝑐𝐴))
31	1, 2, 3, 4, 5, 6, 12, 18, 29, 30	fsumdvdsmul 27121	. 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 27068	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ) → (𝐴 σ 𝑀) = Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} (𝑗↑_𝑐𝐴))
33	1, 32	syldan 591	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝐴 σ 𝑀) = Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} (𝑗↑_𝑐𝐴))
34		sgmval 27068	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴 σ 𝑁) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} (𝑘↑_𝑐𝐴))
35	2, 34	syldan 591	. . 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 7371	. 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 12199	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝑀 gcd 𝑁) = 1)) → (𝑀 · 𝑁) ∈ ℕ)
38		sgmval 27068	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 · 𝑁) ∈ ℕ) → (𝐴 σ (𝑀 · 𝑁)) = Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} (𝑖↑_𝑐𝐴))
39	37, 38	syldan 591	. 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 2775	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 1086 = wceq 1540 ∈ wcel 2109 {crab 3396 class class class wbr 5095 (class class class)co 7353 ℂcc 11026 1c1 11029 · cmul 11033 ℕcn 12146 Σcsu 15611 ∥ cdvds 16181 gcd cgcd 16423 ↑_𝑐ccxp 26480 σ csgm 27022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-pi 15997 df-dvds 16182 df-gcd 16424 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-limc 25783 df-dv 25784 df-log 26481 df-cxp 26482 df-sgm 27028

This theorem is referenced by: perfect1 27155 perfectlem1 27156 perfectlem2 27157 perfectALTVlem1 47706 perfectALTVlem2 47707

W3C validator