Theorem vmaval 27171

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 16711	. . . . 5 ⊢ ℙ ∈ V
2	1	rabex 5345	. . . 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 5152	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴))
6	5	rabbidv 3441	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})
7		vmaval.1	. . . . . . 7 ⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}
8	6, 7	eqtr4di 2793	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = 𝑆)
9	4, 8	sylan9eqr 2797	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → 𝑠 = 𝑆)
10	9	fveqeq2d 6915	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ((♯‘𝑠) = 1 ↔ (♯‘𝑆) = 1))
11	9	unieqd 4925	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ∪ 𝑠 = ∪ 𝑆)
12	11	fveq2d 6911	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (log‘∪ 𝑠) = (log‘∪ 𝑆))
13	10, 12	ifbieq1d 4555	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))
14	3, 13	csbied 3946	. 2 ⊢ (𝑥 = 𝐴 → ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))
15		df-vma 27156	. 2 ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
16		fvex 6920	. . 3 ⊢ (log‘∪ 𝑆) ∈ V
17		c0ex 11253	. . 3 ⊢ 0 ∈ V
18	16, 17	ifex 4581	. 2 ⊢ if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0) ∈ V
19	14, 15, 18	fvmpt 7016	1 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478  ⦋csb 3908  ifcif 4531  ∪ cuni 4912   class class class wbr 5148  ‘cfv 6563  0cc0 11153  1c1 11154  ℕcn 12264  ♯chash 14366   ∥ cdvds 16287  ℙcprime 16705  logclog 26611  Λcvma 27150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-i2m1 11221

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-prm 16706  df-vma 27156

This theorem is referenced by:  isppw  27172  vmappw  27174