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Theorem vmaval 25617
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 16009 . . . . 5 ℙ ∈ V
21rabex 5226 . . . 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 5061 . . . . . . . 8 (𝑥 = 𝐴 → (𝑝𝑥𝑝𝐴))
65rabbidv 3478 . . . . . . 7 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = {𝑝 ∈ ℙ ∣ 𝑝𝐴})
7 vmaval.1 . . . . . . 7 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝𝐴}
86, 7syl6eqr 2871 . . . . . 6 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = 𝑆)
94, 8sylan9eqr 2875 . . . . 5 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → 𝑠 = 𝑆)
109fveqeq2d 6671 . . . 4 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → ((♯‘𝑠) = 1 ↔ (♯‘𝑆) = 1))
119unieqd 4840 . . . . 5 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → 𝑠 = 𝑆)
1211fveq2d 6667 . . . 4 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → (log‘ 𝑠) = (log‘ 𝑆))
1310, 12ifbieq1d 4486 . . 3 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → if((♯‘𝑠) = 1, (log‘ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
143, 13csbied 3916 . 2 (𝑥 = 𝐴{𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
15 df-vma 25602 . 2 Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0))
16 fvex 6676 . . 3 (log‘ 𝑆) ∈ V
17 c0ex 10623 . . 3 0 ∈ V
1816, 17ifex 4511 . 2 if((♯‘𝑆) = 1, (log‘ 𝑆), 0) ∈ V
1914, 15, 18fvmpt 6761 1 (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘ 𝑆), 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  csb 3880  ifcif 4463   cuni 4830   class class class wbr 5057  cfv 6348  0cc0 10525  1c1 10526  cn 11626  chash 13678  cdvds 15595  cprime 16003  logclog 25065  Λcvma 25596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-nn 11627  df-prm 16004  df-vma 25602
This theorem is referenced by:  isppw  25618  vmappw  25620
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