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Theorem pcval 15769
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 473 . . . . . 6 ((𝑝 = 𝑃𝑟 = 𝑁) → 𝑟 = 𝑁)
21eqeq1d 2815 . . . . 5 ((𝑝 = 𝑃𝑟 = 𝑁) → (𝑟 = 0 ↔ 𝑁 = 0))
3 eqeq1 2817 . . . . . . . 8 (𝑟 = 𝑁 → (𝑟 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
4 oveq1 6884 . . . . . . . . . . . . . 14 (𝑝 = 𝑃 → (𝑝𝑛) = (𝑃𝑛))
54breq1d 4861 . . . . . . . . . . . . 13 (𝑝 = 𝑃 → ((𝑝𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝑥))
65rabbidv 3386 . . . . . . . . . . . 12 (𝑝 = 𝑃 → {𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥})
76supeq1d 8594 . . . . . . . . . . 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, 8syl6eqr 2865 . . . . . . . . . 10 (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆)
104breq1d 4861 . . . . . . . . . . . . 13 (𝑝 = 𝑃 → ((𝑝𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 𝑦))
1110rabbidv 3386 . . . . . . . . . . . 12 (𝑝 = 𝑃 → {𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦})
1211supeq1d 8594 . . . . . . . . . . 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, 13syl6eqr 2865 . . . . . . . . . 10 (𝑝 = 𝑃 → sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < ) = 𝑇)
159, 14oveq12d 6895 . . . . . . . . 9 (𝑝 = 𝑃 → (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆𝑇))
1615eqeq2d 2823 . . . . . . . 8 (𝑝 = 𝑃 → (𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑝𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑧 = (𝑆𝑇)))
173, 16bi2anan9r 623 . . . . . . 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 3252 . . . . . 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 6088 . . . . 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 4311 . . . 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 15762 . . . 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 10381 . . . . 5 +∞ ∈ V
23 iotaex 6084 . . . . 5 (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∈ V
2422, 23ifex 4334 . . . 4 if(𝑁 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))) ∈ V
2520, 21, 24ovmpt2a 7024 . . 3 ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) → (𝑃 pCnt 𝑁) = if(𝑁 = 0, +∞, (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))))
26 ifnefalse 4298 . . 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 2867 . 2 (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℚ) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
2827anasss 454 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 384   = wceq 1637  wcel 2157  wne 2985  wrex 3104  {crab 3107  ifcif 4286   class class class wbr 4851  cio 6065  (class class class)co 6877  supcsup 8588  cr 10223  0cc0 10224  +∞cpnf 10359   < clt 10362  cmin 10554   / cdiv 10972  cn 11308  0cn0 11562  cz 11646  cq 12010  cexp 13086  cdvds 15206  cprime 15606   pCnt cpc 15761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-iota 6067  df-fun 6106  df-fv 6112  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-sup 8590  df-pnf 10364  df-xr 10366  df-pc 15762
This theorem is referenced by:  pczpre  15772  pcdiv  15777
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