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Theorem ppival 24653
 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 6557 . . . 4 (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴))
21ineq1d 3775 . . 3 (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ))
32fveq2d 6107 . 2 (𝑥 = 𝐴 → (#‘((0[,]𝑥) ∩ ℙ)) = (#‘((0[,]𝐴) ∩ ℙ)))
4 df-ppi 24626 . 2 π = (𝑥 ∈ ℝ ↦ (#‘((0[,]𝑥) ∩ ℙ)))
5 fvex 6113 . 2 (#‘((0[,]𝐴) ∩ ℙ)) ∈ V
63, 4, 5fvmpt 6191 1 (𝐴 ∈ ℝ → (π𝐴) = (#‘((0[,]𝐴) ∩ ℙ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  [,]cicc 12049  #chash 12979  ℙcprime 15223  πcppi 24620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-ppi 24626 This theorem is referenced by:  ppival2  24654  ppival2g  24655  ppifl  24686  ppiwordi  24688  chtleppi  24735
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