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Theorem vmaval 27141
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 prmex 16678 . . . . 5 ℙ ∈ V
21rabex 5339 . . . 4 {𝑝 ∈ ℙ ∣ 𝑝𝑥} ∈ V
32a1i 11 . . 3 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} ∈ V)
4 id 22 . . . . . 6 (𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥} → 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥})
5 breq2 5157 . . . . . . . 8 (𝑥 = 𝐴 → (𝑝𝑥𝑝𝐴))
65rabbidv 3427 . . . . . . 7 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = {𝑝 ∈ ℙ ∣ 𝑝𝐴})
7 vmaval.1 . . . . . . 7 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝𝐴}
86, 7eqtr4di 2784 . . . . . 6 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = 𝑆)
94, 8sylan9eqr 2788 . . . . 5 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → 𝑠 = 𝑆)
109fveqeq2d 6909 . . . 4 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → ((♯‘𝑠) = 1 ↔ (♯‘𝑆) = 1))
119unieqd 4926 . . . . 5 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → 𝑠 = 𝑆)
1211fveq2d 6905 . . . 4 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → (log‘ 𝑠) = (log‘ 𝑆))
1310, 12ifbieq1d 4557 . . 3 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → if((♯‘𝑠) = 1, (log‘ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
143, 13csbied 3930 . 2 (𝑥 = 𝐴{𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
15 df-vma 27126 . 2 Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0))
16 fvex 6914 . . 3 (log‘ 𝑆) ∈ V
17 c0ex 11258 . . 3 0 ∈ V
1816, 17ifex 4583 . 2 if((♯‘𝑆) = 1, (log‘ 𝑆), 0) ∈ V
1914, 15, 18fvmpt 7009 1 (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  csb 3892  ifcif 4533   cuni 4913   class class class wbr 5153  cfv 6554  0cc0 11158  1c1 11159  cn 12264  chash 14347  cdvds 16256  cprime 16672  logclog 26581  Λcvma 27120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-mulcl 11220  ax-i2m1 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-nn 12265  df-prm 16673  df-vma 27126
This theorem is referenced by:  isppw  27142  vmappw  27144
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