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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 7413 | . . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴)) | |
2 | 1 | ineq1d 4210 | . . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | sumeq1d 15643 | . 2 ⊢ (𝑥 = 𝐴 → Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
4 | df-cht 26590 | . 2 ⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝)) | |
5 | sumex 15630 | . 2 ⊢ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) ∈ V | |
6 | 3, 4, 5 | fvmpt 6995 | 1 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 [,]cicc 13323 Σcsu 15628 ℙcprime 16604 logclog 26054 θccht 26584 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-iota 6492 df-fun 6542 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-seq 13963 df-sum 15629 df-cht 26590 |
This theorem is referenced by: efchtcl 26604 chtge0 26605 chtfl 26642 chtprm 26646 chtnprm 26647 chtwordi 26649 chtdif 26651 cht1 26658 prmorcht 26671 chtlepsi 26698 chtleppi 26702 chpchtsum 26711 chpub 26712 chtppilimlem1 26965 chtvalz 33629 |
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