Theorem pcval 16772

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 484	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑟 = 𝑁) → 𝑟 = 𝑁)
2	1	eqeq1d 2738	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑟 = 𝑁) → (𝑟 = 0 ↔ 𝑁 = 0))
3		eqeq1 2740	. . . . . . . 8 ⊢ (𝑟 = 𝑁 → (𝑟 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
4		oveq1 7365	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑃 → (𝑝↑𝑛) = (𝑃↑𝑛))
5	4	breq1d 5108	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑃 → ((𝑝↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝑥))
6	5	rabbidv 3406	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥})
7	6	supeq1d 9349	. . . . . . . . . . 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 2789	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
10	4	breq1d 5108	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑃 → ((𝑝↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 𝑦))
11	10	rabbidv 3406	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → {𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦})
12	11	supeq1d 9349	. . . . . . . . . . 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 2789	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < ) = 𝑇)
15	9, 14	oveq12d 7376	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − 𝑇))
16	15	eqeq2d 2747	. . . . . . . 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 639	. . . . . . 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 3201	. . . . . 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 6476	. . . . 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 4506	. . . 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 16765	. . . 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 11185	. . . . 5 ⊢ +∞ ∈ V
23		iotaex 6468	. . . . 5 ⊢ (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∈ V
24	22, 23	ifex 4530	. . . 4 ⊢ if(𝑁 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))) ∈ V
25	20, 21, 24	ovmpoa 7513	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) → (𝑃 pCnt 𝑁) = if(𝑁 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))))
26		ifnefalse 4491	. . 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 2791	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
28	27	anasss 466	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060  {crab 3399  ifcif 4479   class class class wbr 5098  ℩cio 6446  (class class class)co 7358  supcsup 9343  ℝcr 11025  0cc0 11026  +∞cpnf 11163   < clt 11166   − cmin 11364   / cdiv 11794  ℕcn 12145  ℕ₀cn0 12401  ℤcz 12488  ℚcq 12861  ↑cexp 13984   ∥ cdvds 16179  ℙcprime 16598   pCnt cpc 16764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-sup 9345  df-pnf 11168  df-pc 16765

This theorem is referenced by:  pczpre  16775  pcdiv  16780