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Theorem ppival 26460
Description: Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
ppival (𝐴 ∈ ℝ → (π𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))

Proof of Theorem ppival
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7361 . . . 4 (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴))
21ineq1d 4169 . . 3 (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ))
32fveq2d 6843 . 2 (𝑥 = 𝐴 → (♯‘((0[,]𝑥) ∩ ℙ)) = (♯‘((0[,]𝐴) ∩ ℙ)))
4 df-ppi 26433 . 2 π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
5 fvex 6852 . 2 (♯‘((0[,]𝐴) ∩ ℙ)) ∈ V
63, 4, 5fvmpt 6945 1 (𝐴 ∈ ℝ → (π𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cin 3907  cfv 6493  (class class class)co 7353  cr 11046  0cc0 11047  [,]cicc 13259  chash 14222  cprime 16539  πcppi 26427
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-ppi 26433
This theorem is referenced by:  ppival2  26461  ppival2g  26462  ppifl  26493  ppiwordi  26495  chtleppi  26542
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