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Mirrors > Home > MPE Home > Th. List > cht1 | Structured version Visualization version GIF version |
Description: The Chebyshev function at 1. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
cht1 | ⊢ (θ‘1) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10077 | . . 3 ⊢ 1 ∈ ℝ | |
2 | chtval 24881 | . . 3 ⊢ (1 ∈ ℝ → (θ‘1) = Σ𝑝 ∈ ((0[,]1) ∩ ℙ)(log‘𝑝)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (θ‘1) = Σ𝑝 ∈ ((0[,]1) ∩ ℙ)(log‘𝑝) |
4 | ppisval 24875 | . . . . 5 ⊢ (1 ∈ ℝ → ((0[,]1) ∩ ℙ) = ((2...(⌊‘1)) ∩ ℙ)) | |
5 | 1, 4 | ax-mp 5 | . . . 4 ⊢ ((0[,]1) ∩ ℙ) = ((2...(⌊‘1)) ∩ ℙ) |
6 | 1z 11445 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
7 | flid 12649 | . . . . . . . 8 ⊢ (1 ∈ ℤ → (⌊‘1) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ (⌊‘1) = 1 |
9 | 8 | oveq2i 6701 | . . . . . 6 ⊢ (2...(⌊‘1)) = (2...1) |
10 | 1lt2 11232 | . . . . . . 7 ⊢ 1 < 2 | |
11 | 2z 11447 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
12 | fzn 12395 | . . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ 1 ∈ ℤ) → (1 < 2 ↔ (2...1) = ∅)) | |
13 | 11, 6, 12 | mp2an 708 | . . . . . . 7 ⊢ (1 < 2 ↔ (2...1) = ∅) |
14 | 10, 13 | mpbi 220 | . . . . . 6 ⊢ (2...1) = ∅ |
15 | 9, 14 | eqtri 2673 | . . . . 5 ⊢ (2...(⌊‘1)) = ∅ |
16 | 15 | ineq1i 3843 | . . . 4 ⊢ ((2...(⌊‘1)) ∩ ℙ) = (∅ ∩ ℙ) |
17 | 0in 4002 | . . . 4 ⊢ (∅ ∩ ℙ) = ∅ | |
18 | 5, 16, 17 | 3eqtri 2677 | . . 3 ⊢ ((0[,]1) ∩ ℙ) = ∅ |
19 | 18 | sumeq1i 14472 | . 2 ⊢ Σ𝑝 ∈ ((0[,]1) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ∅ (log‘𝑝) |
20 | sum0 14496 | . 2 ⊢ Σ𝑝 ∈ ∅ (log‘𝑝) = 0 | |
21 | 3, 19, 20 | 3eqtri 2677 | 1 ⊢ (θ‘1) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ∈ wcel 2030 ∩ cin 3606 ∅c0 3948 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 < clt 10112 2c2 11108 ℤcz 11415 [,]cicc 12216 ...cfz 12364 ⌊cfl 12631 Σcsu 14460 ℙcprime 15432 logclog 24346 θccht 24862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-sum 14461 df-dvds 15028 df-prm 15433 df-cht 24868 |
This theorem is referenced by: cht2 24943 |
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