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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 7167 | . . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴)) | |
2 | 1 | ineq1d 4191 | . . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | sumeq1d 15061 | . 2 ⊢ (𝑥 = 𝐴 → Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
4 | df-cht 25677 | . 2 ⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝)) | |
5 | sumex 15047 | . 2 ⊢ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) ∈ V | |
6 | 3, 4, 5 | fvmpt 6771 | 1 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 0cc0 10540 [,]cicc 12744 Σcsu 15045 ℙcprime 16018 logclog 25141 θccht 25671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-iota 6317 df-fun 6360 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-seq 13373 df-sum 15046 df-cht 25677 |
This theorem is referenced by: efchtcl 25691 chtge0 25692 chtfl 25729 chtprm 25733 chtnprm 25734 chtwordi 25736 chtdif 25738 cht1 25745 prmorcht 25758 chtlepsi 25785 chtleppi 25789 chpchtsum 25798 chpub 25799 chtppilimlem1 26052 chtvalz 31904 |
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