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Mirrors > Home > MPE Home > Th. List > sgmval | Structured version Visualization version GIF version |
Description: The value of the divisor function. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 21-Jun-2015.) |
Ref | Expression |
---|---|
sgmval | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝑐𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑛 = 𝐵) → 𝑛 = 𝐵) | |
2 | 1 | breq2d 5153 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑛 = 𝐵) → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝐵)) |
3 | 2 | rabbidv 3434 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑛 = 𝐵) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} = {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵}) |
4 | simpll 764 | . . . 4 ⊢ (((𝑥 = 𝐴 ∧ 𝑛 = 𝐵) ∧ 𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛}) → 𝑥 = 𝐴) | |
5 | 4 | oveq2d 7420 | . . 3 ⊢ (((𝑥 = 𝐴 ∧ 𝑛 = 𝐵) ∧ 𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛}) → (𝑘↑𝑐𝑥) = (𝑘↑𝑐𝐴)) |
6 | 3, 5 | sumeq12dv 15656 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑛 = 𝐵) → Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝑐𝐴)) |
7 | df-sgm 26985 | . 2 ⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥)) | |
8 | sumex 15638 | . 2 ⊢ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝑐𝐴) ∈ V | |
9 | 6, 7, 8 | ovmpoa 7558 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ) → (𝐴 σ 𝐵) = Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} (𝑘↑𝑐𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 class class class wbr 5141 (class class class)co 7404 ℂcc 11107 ℕcn 12213 Σcsu 15636 ∥ cdvds 16202 ↑𝑐ccxp 26440 σ csgm 26979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-seq 13970 df-sum 15637 df-sgm 26985 |
This theorem is referenced by: sgmval2 27026 sgmppw 27081 sgmmul 27085 perfectlem2 27114 perfectALTVlem2 46943 |
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