Theorem vmaval 27030

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 16654	. . . . 5 ⊢ ℙ ∈ V
2	1	rabex 5297	. . . 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 5114	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴))
6	5	rabbidv 3416	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})
7		vmaval.1	. . . . . . 7 ⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}
8	6, 7	eqtr4di 2783	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = 𝑆)
9	4, 8	sylan9eqr 2787	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → 𝑠 = 𝑆)
10	9	fveqeq2d 6869	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ((♯‘𝑠) = 1 ↔ (♯‘𝑆) = 1))
11	9	unieqd 4887	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ∪ 𝑠 = ∪ 𝑆)
12	11	fveq2d 6865	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (log‘∪ 𝑠) = (log‘∪ 𝑆))
13	10, 12	ifbieq1d 4516	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))
14	3, 13	csbied 3901	. 2 ⊢ (𝑥 = 𝐴 → ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))
15		df-vma 27015	. 2 ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
16		fvex 6874	. . 3 ⊢ (log‘∪ 𝑆) ∈ V
17		c0ex 11175	. . 3 ⊢ 0 ∈ V
18	16, 17	ifex 4542	. 2 ⊢ if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0) ∈ V
19	14, 15, 18	fvmpt 6971	1 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450  ⦋csb 3865  ifcif 4491  ∪ cuni 4874   class class class wbr 5110  ‘cfv 6514  0cc0 11075  1c1 11076  ℕcn 12193  ♯chash 14302   ∥ cdvds 16229  ℙcprime 16648  logclog 26470  Λcvma 27009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-mulcl 11137  ax-i2m1 11143

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-nn 12194  df-prm 16649  df-vma 27015

This theorem is referenced by:  isppw  27031  vmappw  27033