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 ∣ 𝑝 ∥ 𝐴}

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.

Step	Hyp	Ref	Expression
1		nnex 10986	. . . . . 6 ⊢ ℕ ∈ V
2		prmnn 15331	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
3	2	ssriv 3592	. . . . . 6 ⊢ ℙ ⊆ ℕ
4	1, 3	ssexi 4773	. . . . 5 ⊢ ℙ ∈ V
5	4	rabex 4783	. . . 4 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} ∈ V
6	5	a1i 11	. . 3 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} ∈ V)
7		id 22	. . . . . . 7 ⊢ (𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} → 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥})
8		breq2 4627	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴))
9	8	rabbidv 3181	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})
10		vmaval.1	. . . . . . . 8 ⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}
11	9, 10	syl6eqr 2673	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = 𝑆)
12	7, 11	sylan9eqr 2677	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → 𝑠 = 𝑆)
13	12	fveq2d 6162	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (#‘𝑠) = (#‘𝑆))
14	13	eqeq1d 2623	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ((#‘𝑠) = 1 ↔ (#‘𝑆) = 1))
15	12	unieqd 4419	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ∪ 𝑠 = ∪ 𝑆)
16	15	fveq2d 6162	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (log‘∪ 𝑠) = (log‘∪ 𝑆))
17	14, 16	ifbieq1d 4087	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → if((#‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0))
18	6, 17	csbied 3546	. 2 ⊢ (𝑥 = 𝐴 → ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0))
19		df-vma 24758	. 2 ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0))
20		fvex 6168	. . 3 ⊢ (log‘∪ 𝑆) ∈ V
21		c0ex 9994	. . 3 ⊢ 0 ∈ V
22	20, 21	ifex 4134	. 2 ⊢ if((#‘𝑆) = 1, (log‘∪ 𝑆), 0) ∈ V
23	18, 19, 22	fvmpt 6249	1 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0))

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  ℕcn 10980  #chash 13073   ∥ cdvds 14926  ℙcprime 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