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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 7143 | . . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴)) | |
2 | 1 | ineq1d 4138 | . . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | sumeq1d 15050 | . 2 ⊢ (𝑥 = 𝐴 → Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
4 | df-cht 25682 | . 2 ⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝)) | |
5 | sumex 15036 | . 2 ⊢ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) ∈ V | |
6 | 3, 4, 5 | fvmpt 6745 | 1 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 [,]cicc 12729 Σcsu 15034 ℙcprime 16005 logclog 25146 θccht 25676 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-iota 6283 df-fun 6326 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seq 13365 df-sum 15035 df-cht 25682 |
This theorem is referenced by: efchtcl 25696 chtge0 25697 chtfl 25734 chtprm 25738 chtnprm 25739 chtwordi 25741 chtdif 25743 cht1 25750 prmorcht 25763 chtlepsi 25790 chtleppi 25794 chpchtsum 25803 chpub 25804 chtppilimlem1 26057 chtvalz 32010 |
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