MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ppival Structured version   Visualization version   GIF version

Theorem ppival 26382
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 7345 . . . 4 (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴))
21ineq1d 4158 . . 3 (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ))
32fveq2d 6829 . 2 (𝑥 = 𝐴 → (♯‘((0[,]𝑥) ∩ ℙ)) = (♯‘((0[,]𝐴) ∩ ℙ)))
4 df-ppi 26355 . 2 π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ)))
5 fvex 6838 . 2 (♯‘((0[,]𝐴) ∩ ℙ)) ∈ V
63, 4, 5fvmpt 6931 1 (𝐴 ∈ ℝ → (π𝐴) = (♯‘((0[,]𝐴) ∩ ℙ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  cin 3897  cfv 6479  (class class class)co 7337  cr 10971  0cc0 10972  [,]cicc 13183  chash 14145  cprime 16473  πcppi 26349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-ov 7340  df-ppi 26355
This theorem is referenced by:  ppival2  26383  ppival2g  26384  ppifl  26415  ppiwordi  26417  chtleppi  26464
  Copyright terms: Public domain W3C validator