Theorem pcval 17028

Description: The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Hypotheses

Ref	Expression
pcval.1	⊢ 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
pcval.2	⊢ 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )

Assertion

Ref	Expression
pcval	⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))

Distinct variable groups:   𝑥,𝑛,𝑦,𝑧,𝑁   𝑃,𝑛,𝑥,𝑦,𝑧   𝑧,𝑆   𝑧,𝑇

Allowed substitution hints:   𝑆(𝑥, 𝑦, 𝑛)   𝑇(𝑥, 𝑦, 𝑛)

Proof of Theorem pcval

Dummy variables 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑟 = 𝑁) → 𝑟 = 𝑁)
2	1	eqeq1d 2762	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑟 = 𝑁) → (𝑟 = 0 ↔ 𝑁 = 0))
3		eqeq1 2764	. . . . . . . 8 ⊢ (𝑟 = 𝑁 → (𝑟 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
4		oveq1 7428	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑃 → (𝑝↑𝑛) = (𝑃↑𝑛))
5	4	breq1d 5112	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑃 → ((𝑝↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝑥))
6	5	rabbidv 3419	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥})
7	6	supeq1d 9442	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ))
8		pcval.1	. . . . . . . . . . 11 ⊢ 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
9	7, 8	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
10	4	breq1d 5112	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑃 → ((𝑝↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 𝑦))
11	10	rabbidv 3419	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦})
12	11	supeq1d 9442	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))
13		pcval.2	. . . . . . . . . . 11 ⊢ 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
14	12, 13	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < ) = 𝑇)
15	9, 14	oveq12d 7439	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − 𝑇))
16	15	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑧 = (𝑆 − 𝑇)))
17	3, 16	bi2anan9r 651	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑟 = 𝑁) → ((𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
18	17	2rexbidv 3227	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑟 = 𝑁) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
19	18	iotabidv 6522	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑟 = 𝑁) → (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < )))) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
20	2, 19	ifbieq2d 4508	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑟 = 𝑁) → if(𝑟 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < ))))) = if(𝑁 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))))
21		df-pc 17021	. . . 4 ⊢ pCnt = (𝑝 ∈ ℙ, 𝑟 ∈ ℚ ↦ if(𝑟 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < ))))))
22		pnfex 11366	. . . . 5 ⊢ +∞ ∈ V
23		iotaex 6514	. . . . 5 ⊢ (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∈ V
24	22, 23	ifex 4532	. . . 4 ⊢ if(𝑁 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))) ∈ V
25	20, 21, 24	ovmpoa 7576	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) → (𝑃 pCnt 𝑁) = if(𝑁 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))))
26		ifnefalse 4493	. . 3 ⊢ (𝑁 ≠ 0 → if(𝑁 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
27	25, 26	sylan9eq 2815	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
28	27	anasss 472	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  {crab 3412  ifcif 4481   class class class wbr 5102  ℩cio 6492  (class class class)co 7421  supcsup 9436  ℝcr 11203  0cc0 11204  +∞cpnf 11344   < clt 11347   − cmin 11545   / cdiv 11977  ℕcn 12339  ℕ₀cn0 12610  ℤcz 12697  ℚcq 13079  ↑cexp 14208   ∥ cdvds 16428  ℙcprime 16852   pCnt cpc 17020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-sup 9438  df-pnf 11349  df-pc 17021

This theorem is used by:  pczpre  17031  pcdiv  17036