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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 7199 | . . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴)) | |
2 | 1 | ineq1d 4112 | . . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | fveq2d 6699 | . 2 ⊢ (𝑥 = 𝐴 → (♯‘((0[,]𝑥) ∩ ℙ)) = (♯‘((0[,]𝐴) ∩ ℙ))) |
4 | df-ppi 25936 | . 2 ⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ))) | |
5 | fvex 6708 | . 2 ⊢ (♯‘((0[,]𝐴) ∩ ℙ)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6796 | 1 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 0cc0 10694 [,]cicc 12903 ♯chash 13861 ℙcprime 16191 πcppi 25930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-ppi 25936 |
This theorem is referenced by: ppival2 25964 ppival2g 25965 ppifl 25996 ppiwordi 25998 chtleppi 26045 |
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