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Mirrors > Home > MPE Home > Th. List > ppival | Structured version Visualization version GIF version |
Description: Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
ppival | ⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7345 | . . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴)) | |
2 | 1 | ineq1d 4158 | . . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | fveq2d 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 | |
6 | 3, 4, 5 | fvmpt 6931 | 1 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (♯‘((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 ℝcr 10971 0cc0 10972 [,]cicc 13183 ♯chash 14145 ℙcprime 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 |
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