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Theorem ppival 25073
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 6800 . . . 4 (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴))
21ineq1d 3962 . . 3 (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ))
32fveq2d 6336 . 2 (𝑥 = 𝐴 → (♯‘((0[,]𝑥) ∩ ℙ)) = (♯‘((0[,]𝐴) ∩ ℙ)))
4 df-ppi 25046 . 2 π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
5 fvex 6342 . 2 (♯‘((0[,]𝐴) ∩ ℙ)) ∈ V
63, 4, 5fvmpt 6424 1 (𝐴 ∈ ℝ → (π𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  cin 3720  cfv 6031  (class class class)co 6792  cr 10136  0cc0 10137  [,]cicc 12382  chash 13320  cprime 15591  πcppi 25040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-ppi 25046
This theorem is referenced by:  ppival2  25074  ppival2g  25075  ppifl  25106  ppiwordi  25108  chtleppi  25155
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