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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 7164 | . . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴)) | |
2 | 1 | ineq1d 4188 | . . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | fveq2d 6674 | . 2 ⊢ (𝑥 = 𝐴 → (♯‘((0[,]𝑥) ∩ ℙ)) = (♯‘((0[,]𝐴) ∩ ℙ))) |
4 | df-ppi 25677 | . 2 ⊢ π = (𝑥 ∈ ℝ ↦ (♯‘((0[,]𝑥) ∩ ℙ))) | |
5 | fvex 6683 | . 2 ⊢ (♯‘((0[,]𝐴) ∩ ℙ)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6768 | 1 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (♯‘((0[,]𝐴) ∩ ℙ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 [,]cicc 12742 ♯chash 13691 ℙcprime 16015 πcppi 25671 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-ppi 25677 |
This theorem is referenced by: ppival2 25705 ppival2g 25706 ppifl 25737 ppiwordi 25739 chtleppi 25786 |
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