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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  cn 11222  chash 13317  cdvds 15185  cprime 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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