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Theorem pcval 16175
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 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
pcval.2 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
Assertion
Ref Expression
pcval ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
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.
StepHypRef Expression
1 simpr 488 . . . . . 6 ((𝑝 = 𝑃𝑟 = 𝑁) → 𝑟 = 𝑁)
21eqeq1d 2803 . . . . 5 ((𝑝 = 𝑃𝑟 = 𝑁) → (𝑟 = 0 ↔ 𝑁 = 0))
3 eqeq1 2805 . . . . . . . 8 (𝑟 = 𝑁 → (𝑟 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
4 oveq1 7146 . . . . . . . . . . . . . 14 (𝑝 = 𝑃 → (𝑝𝑛) = (𝑃𝑛))
54breq1d 5043 . . . . . . . . . . . . 13 (𝑝 = 𝑃 → ((𝑝𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝑥))
65rabbidv 3430 . . . . . . . . . . . 12 (𝑝 = 𝑃 → {𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥})
76supeq1d 8898 . . . . . . . . . . 11 (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ))
8 pcval.1 . . . . . . . . . . 11 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
97, 8eqtr4di 2854 . . . . . . . . . 10 (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
104breq1d 5043 . . . . . . . . . . . . 13 (𝑝 = 𝑃 → ((𝑝𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 𝑦))
1110rabbidv 3430 . . . . . . . . . . . 12 (𝑝 = 𝑃 → {𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦})
1211supeq1d 8898 . . . . . . . . . . 11 (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))
13 pcval.2 . . . . . . . . . . 11 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
1412, 13eqtr4di 2854 . . . . . . . . . 10 (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < ) = 𝑇)
159, 14oveq12d 7157 . . . . . . . . 9 (𝑝 = 𝑃 → (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆𝑇))
1615eqeq2d 2812 . . . . . . . 8 (𝑝 = 𝑃 → (𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑧 = (𝑆𝑇)))
173, 16bi2anan9r 639 . . . . . . 7 ((𝑝 = 𝑃𝑟 = 𝑁) → ((𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
18172rexbidv 3262 . . . . . 6 ((𝑝 = 𝑃𝑟 = 𝑁) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
1918iotabidv 6312 . . . . 5 ((𝑝 = 𝑃𝑟 = 𝑁) → (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < )))) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
202, 19ifbieq2d 4453 . . . 4 ((𝑝 = 𝑃𝑟 = 𝑁) → if(𝑟 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < ))))) = if(𝑁 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))))
21 df-pc 16168 . . . 4 pCnt = (𝑝 ∈ ℙ, 𝑟 ∈ ℚ ↦ if(𝑟 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < ))))))
22 pnfex 10687 . . . . 5 +∞ ∈ V
23 iotaex 6308 . . . . 5 (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∈ V
2422, 23ifex 4476 . . . 4 if(𝑁 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))) ∈ V
2520, 21, 24ovmpoa 7288 . . 3 ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) → (𝑃 pCnt 𝑁) = if(𝑁 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))))
26 ifnefalse 4440 . . 3 (𝑁 ≠ 0 → if(𝑁 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
2725, 26sylan9eq 2856 . 2 (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
2827anasss 470 1 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  wrex 3110  {crab 3113  ifcif 4428   class class class wbr 5033  cio 6285  (class class class)co 7139  supcsup 8892  cr 10529  0cc0 10530  +∞cpnf 10665   < clt 10668  cmin 10863   / cdiv 11290  cn 11629  0cn0 11889  cz 11973  cq 12340  cexp 13429  cdvds 15603  cprime 16009   pCnt cpc 16167
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-sup 8894  df-pnf 10670  df-pc 16168
This theorem is referenced by:  pczpre  16178  pcdiv  16183
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