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Theorem vmaval 25056
 Description: Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Hypothesis
Ref Expression
vmaval.1 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝𝐴}
Assertion
Ref Expression
vmaval (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
Distinct variable group:   𝐴,𝑝
Allowed substitution hint:   𝑆(𝑝)

Proof of Theorem vmaval
Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 11228 . . . . . 6 ℕ ∈ V
2 prmnn 15591 . . . . . . 7 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
32ssriv 3756 . . . . . 6 ℙ ⊆ ℕ
41, 3ssexi 4937 . . . . 5 ℙ ∈ V
54rabex 4946 . . . 4 {𝑝 ∈ ℙ ∣ 𝑝𝑥} ∈ V
65a1i 11 . . 3 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} ∈ V)
7 id 22 . . . . . . 7 (𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥} → 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥})
8 breq2 4790 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑝𝑥𝑝𝐴))
98rabbidv 3339 . . . . . . . 8 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = {𝑝 ∈ ℙ ∣ 𝑝𝐴})
10 vmaval.1 . . . . . . . 8 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝𝐴}
119, 10syl6eqr 2823 . . . . . . 7 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = 𝑆)
127, 11sylan9eqr 2827 . . . . . 6 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → 𝑠 = 𝑆)
1312fveq2d 6334 . . . . 5 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → (♯‘𝑠) = (♯‘𝑆))
1413eqeq1d 2773 . . . 4 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → ((♯‘𝑠) = 1 ↔ (♯‘𝑆) = 1))
1512unieqd 4584 . . . . 5 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → 𝑠 = 𝑆)
1615fveq2d 6334 . . . 4 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → (log‘ 𝑠) = (log‘ 𝑆))
1714, 16ifbieq1d 4248 . . 3 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → if((♯‘𝑠) = 1, (log‘ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
186, 17csbied 3709 . 2 (𝑥 = 𝐴{𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
19 df-vma 25041 . 2 Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0))
20 fvex 6341 . . 3 (log‘ 𝑆) ∈ V
21 c0ex 10236 . . 3 0 ∈ V
2220, 21ifex 4295 . 2 if((♯‘𝑆) = 1, (log‘ 𝑆), 0) ∈ V
2318, 19, 22fvmpt 6423 1 (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {crab 3065  Vcvv 3351  ⦋csb 3682  ifcif 4225  ∪ cuni 4574   class class class wbr 4786  ‘cfv 6029  0cc0 10138  1c1 10139  ℕcn 11222  ♯chash 13317   ∥ cdvds 15185  ℙcprime 15588  logclog 24518  Λcvma 25035 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-i2m1 10206  ax-1ne0 10207  ax-rrecex 10210  ax-cnre 10211 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6795  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-nn 11223  df-prm 15589  df-vma 25041 This theorem is referenced by:  isppw  25057  vmappw  25059
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