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Theorem ppival 24753
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 6612 . . . 4 (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴))
21ineq1d 3791 . . 3 (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ))
32fveq2d 6152 . 2 (𝑥 = 𝐴 → (#‘((0[,]𝑥) ∩ ℙ)) = (#‘((0[,]𝐴) ∩ ℙ)))
4 df-ppi 24726 . 2 π = (𝑥 ∈ ℝ ↦ (#‘((0[,]𝑥) ∩ ℙ)))
5 fvex 6158 . 2 (#‘((0[,]𝐴) ∩ ℙ)) ∈ V
63, 4, 5fvmpt 6239 1 (𝐴 ∈ ℝ → (π𝐴) = (#‘((0[,]𝐴) ∩ ℙ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cin 3554  cfv 5847  (class class class)co 6604  cr 9879  0cc0 9880  [,]cicc 12120  #chash 13057  cprime 15309  πcppi 24720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-ppi 24726
This theorem is referenced by:  ppival2  24754  ppival2g  24755  ppifl  24786  ppiwordi  24788  chtleppi  24835
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