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Theorem vmaval 24773
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 10986 . . . . . 6 ℕ ∈ V
2 prmnn 15331 . . . . . . 7 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
32ssriv 3592 . . . . . 6 ℙ ⊆ ℕ
41, 3ssexi 4773 . . . . 5 ℙ ∈ V
54rabex 4783 . . . 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 4627 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑝𝑥𝑝𝐴))
98rabbidv 3181 . . . . . . . 8 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = {𝑝 ∈ ℙ ∣ 𝑝𝐴})
10 vmaval.1 . . . . . . . 8 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝𝐴}
119, 10syl6eqr 2673 . . . . . . 7 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = 𝑆)
127, 11sylan9eqr 2677 . . . . . 6 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → 𝑠 = 𝑆)
1312fveq2d 6162 . . . . 5 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → (#‘𝑠) = (#‘𝑆))
1413eqeq1d 2623 . . . 4 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → ((#‘𝑠) = 1 ↔ (#‘𝑆) = 1))
1512unieqd 4419 . . . . 5 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → 𝑠 = 𝑆)
1615fveq2d 6162 . . . 4 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → (log‘ 𝑠) = (log‘ 𝑆))
1714, 16ifbieq1d 4087 . . 3 ((𝑥 = 𝐴𝑠 = {𝑝 ∈ ℙ ∣ 𝑝𝑥}) → if((#‘𝑠) = 1, (log‘ 𝑠), 0) = if((#‘𝑆) = 1, (log‘ 𝑆), 0))
186, 17csbied 3546 . 2 (𝑥 = 𝐴{𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0) = if((#‘𝑆) = 1, (log‘ 𝑆), 0))
19 df-vma 24758 . 2 Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0))
20 fvex 6168 . . 3 (log‘ 𝑆) ∈ V
21 c0ex 9994 . . 3 0 ∈ V
2220, 21ifex 4134 . 2 if((#‘𝑆) = 1, (log‘ 𝑆), 0) ∈ V
2318, 19, 22fvmpt 6249 1 (𝐴 ∈ ℕ → (Λ‘𝐴) = if((#‘𝑆) = 1, (log‘ 𝑆), 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  csb 3519  ifcif 4064   cuni 4409   class class class wbr 4623  cfv 5857  0cc0 9896  1c1 9897  cn 10980  #chash 13073  cdvds 14926  cprime 15328  logclog 24239  Λcvma 24752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-i2m1 9964  ax-1ne0 9965  ax-rrecex 9968  ax-cnre 9969
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-nn 10981  df-prm 15329  df-vma 24758
This theorem is referenced by:  isppw  24774  vmappw  24776
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