![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7361 | . . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴)) | |
2 | 1 | ineq1d 4169 | . . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | fveq2d 6843 | . 2 ⊢ (𝑥 = 𝐴 → (♯‘((0[,]𝑥) ∩ ℙ)) = (♯‘((0[,]𝐴) ∩ ℙ))) |
4 | df-ppi 26433 | . 2 ⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ))) | |
5 | fvex 6852 | . 2 ⊢ (♯‘((0[,]𝐴) ∩ ℙ)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6945 | 1 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∩ cin 3907 ‘cfv 6493 (class class class)co 7353 ℝcr 11046 0cc0 11047 [,]cicc 13259 ♯chash 14222 ℙcprime 16539 πcppi 26427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-ov 7356 df-ppi 26433 |
This theorem is referenced by: ppival2 26461 ppival2g 26462 ppifl 26493 ppiwordi 26495 chtleppi 26542 |
Copyright terms: Public domain | W3C validator |