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Mirrors > Home > MPE Home > Th. List > chtfl | Structured version Visualization version GIF version |
Description: The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
chtfl | ⊢ (𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flidm 13540 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴)) | |
2 | 1 | oveq2d 7288 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (2...(⌊‘(⌊‘𝐴))) = (2...(⌊‘𝐴))) |
3 | 2 | ineq1d 4151 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((2...(⌊‘(⌊‘𝐴))) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ)) |
4 | reflcl 13527 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
5 | ppisval 26264 | . . . . 5 ⊢ ((⌊‘𝐴) ∈ ℝ → ((0[,](⌊‘𝐴)) ∩ ℙ) = ((2...(⌊‘(⌊‘𝐴))) ∩ ℙ)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,](⌊‘𝐴)) ∩ ℙ) = ((2...(⌊‘(⌊‘𝐴))) ∩ ℙ)) |
7 | ppisval 26264 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ)) | |
8 | 3, 6, 7 | 3eqtr4d 2790 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0[,](⌊‘𝐴)) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ)) |
9 | 8 | sumeq1d 15424 | . 2 ⊢ (𝐴 ∈ ℝ → Σ𝑝 ∈ ((0[,](⌊‘𝐴)) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
10 | chtval 26270 | . . 3 ⊢ ((⌊‘𝐴) ∈ ℝ → (θ‘(⌊‘𝐴)) = Σ𝑝 ∈ ((0[,](⌊‘𝐴)) ∩ ℙ)(log‘𝑝)) | |
11 | 4, 10 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = Σ𝑝 ∈ ((0[,](⌊‘𝐴)) ∩ ℙ)(log‘𝑝)) |
12 | chtval 26270 | . 2 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) | |
13 | 9, 11, 12 | 3eqtr4d 2790 | 1 ⊢ (𝐴 ∈ ℝ → (θ‘(⌊‘𝐴)) = (θ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ∩ cin 3891 ‘cfv 6432 (class class class)co 7272 ℝcr 10881 0cc0 10882 2c2 12039 [,]cicc 13093 ...cfz 13250 ⌊cfl 13521 Σcsu 15408 ℙcprime 16387 logclog 25721 θccht 26251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-sup 9189 df-inf 9190 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-n0 12245 df-z 12331 df-uz 12594 df-rp 12742 df-icc 13097 df-fz 13251 df-fl 13523 df-seq 13733 df-exp 13794 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-sum 15409 df-dvds 15975 df-prm 16388 df-cht 26257 |
This theorem is referenced by: efchtdvds 26319 chtub 26371 |
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