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Mirrors > Home > MPE Home > Th. List > chtval | Structured version Visualization version GIF version |
Description: Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
chtval | ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6557 | . . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴)) | |
2 | 1 | ineq1d 3775 | . . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | sumeq1d 14279 | . 2 ⊢ (𝑥 = 𝐴 → Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
4 | df-cht 24623 | . 2 ⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝)) | |
5 | sumex 14266 | . 2 ⊢ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) ∈ V | |
6 | 3, 4, 5 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 [,]cicc 12049 Σcsu 14264 ℙcprime 15223 logclog 24105 θccht 24617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-sum 14265 df-cht 24623 |
This theorem is referenced by: efchtcl 24637 chtge0 24638 chtfl 24675 chtprm 24679 chtnprm 24680 chtwordi 24682 chtdif 24684 cht1 24691 prmorcht 24704 chtlepsi 24731 chtleppi 24735 chpchtsum 24744 chpub 24745 chtppilimlem1 24962 |
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