Theorem vmaval 25698

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		prmex 16011	. . . . 5 ⊢ ℙ ∈ V
2	1	rabex 5199	. . . 4 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} ∈ V
3	2	a1i 11	. . 3 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} ∈ V)
4		id 22	. . . . . 6 ⊢ (𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} → 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥})
5		breq2 5034	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴))
6	5	rabbidv 3427	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})
7		vmaval.1	. . . . . . 7 ⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}
8	6, 7	eqtr4di 2851	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = 𝑆)
9	4, 8	sylan9eqr 2855	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → 𝑠 = 𝑆)
10	9	fveqeq2d 6653	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ((♯‘𝑠) = 1 ↔ (♯‘𝑆) = 1))
11	9	unieqd 4814	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ∪ 𝑠 = ∪ 𝑆)
12	11	fveq2d 6649	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (log‘∪ 𝑠) = (log‘∪ 𝑆))
13	10, 12	ifbieq1d 4448	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))
14	3, 13	csbied 3864	. 2 ⊢ (𝑥 = 𝐴 → ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))
15		df-vma 25683	. 2 ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
16		fvex 6658	. . 3 ⊢ (log‘∪ 𝑆) ∈ V
17		c0ex 10624	. . 3 ⊢ 0 ∈ V
18	16, 17	ifex 4473	. 2 ⊢ if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0) ∈ V
19	14, 15, 18	fvmpt 6745	1 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441  ⦋csb 3828  ifcif 4425  ∪ cuni 4800   class class class wbr 5030  ‘cfv 6324  0cc0 10526  1c1 10527  ℕcn 11625  ♯chash 13686   ∥ cdvds 15599  ℙcprime 16005  logclog 25146  Λcvma 25677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-mulcl 10588  ax-i2m1 10594

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-nn 11626  df-prm 16006  df-vma 25683

This theorem is referenced by:  isppw  25699  vmappw  25701