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Mirrors > Home > MPE Home > Th. List > cht2 | Structured version Visualization version GIF version |
Description: The Chebyshev function at 2. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
cht2 | ⊢ (θ‘2) = (log‘2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 11375 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | fveq2i 6415 | . 2 ⊢ (θ‘2) = (θ‘(1 + 1)) |
3 | 1z 11696 | . . 3 ⊢ 1 ∈ ℤ | |
4 | 2prm 15738 | . . . 4 ⊢ 2 ∈ ℙ | |
5 | 1, 4 | eqeltrri 2876 | . . 3 ⊢ (1 + 1) ∈ ℙ |
6 | chtprm 25230 | . . 3 ⊢ ((1 ∈ ℤ ∧ (1 + 1) ∈ ℙ) → (θ‘(1 + 1)) = ((θ‘1) + (log‘(1 + 1)))) | |
7 | 3, 5, 6 | mp2an 684 | . 2 ⊢ (θ‘(1 + 1)) = ((θ‘1) + (log‘(1 + 1))) |
8 | cht1 25242 | . . . . 5 ⊢ (θ‘1) = 0 | |
9 | 8 | eqcomi 2809 | . . . 4 ⊢ 0 = (θ‘1) |
10 | 1 | fveq2i 6415 | . . . 4 ⊢ (log‘2) = (log‘(1 + 1)) |
11 | 9, 10 | oveq12i 6891 | . . 3 ⊢ (0 + (log‘2)) = ((θ‘1) + (log‘(1 + 1))) |
12 | 2rp 12078 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
13 | relogcl 24662 | . . . . . 6 ⊢ (2 ∈ ℝ+ → (log‘2) ∈ ℝ) | |
14 | 12, 13 | ax-mp 5 | . . . . 5 ⊢ (log‘2) ∈ ℝ |
15 | 14 | recni 10344 | . . . 4 ⊢ (log‘2) ∈ ℂ |
16 | 15 | addid2i 10515 | . . 3 ⊢ (0 + (log‘2)) = (log‘2) |
17 | 11, 16 | eqtr3i 2824 | . 2 ⊢ ((θ‘1) + (log‘(1 + 1))) = (log‘2) |
18 | 2, 7, 17 | 3eqtri 2826 | 1 ⊢ (θ‘2) = (log‘2) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ‘cfv 6102 (class class class)co 6879 ℝcr 10224 0cc0 10225 1c1 10226 + caddc 10228 2c2 11367 ℤcz 11665 ℝ+crp 12073 ℙcprime 15718 logclog 24641 θccht 25168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-pre-sup 10303 ax-addf 10304 ax-mulf 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-of 7132 df-om 7301 df-1st 7402 df-2nd 7403 df-supp 7534 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-2o 7801 df-oadd 7804 df-er 7983 df-map 8098 df-pm 8099 df-ixp 8150 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-fsupp 8519 df-fi 8560 df-sup 8591 df-inf 8592 df-oi 8658 df-card 9052 df-cda 9279 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-9 11382 df-n0 11580 df-z 11666 df-dec 11783 df-uz 11930 df-q 12033 df-rp 12074 df-xneg 12192 df-xadd 12193 df-xmul 12194 df-ioo 12427 df-ioc 12428 df-ico 12429 df-icc 12430 df-fz 12580 df-fzo 12720 df-fl 12847 df-mod 12923 df-seq 13055 df-exp 13114 df-fac 13313 df-bc 13342 df-hash 13370 df-shft 14147 df-cj 14179 df-re 14180 df-im 14181 df-sqrt 14315 df-abs 14316 df-limsup 14542 df-clim 14559 df-rlim 14560 df-sum 14757 df-ef 15133 df-sin 15135 df-cos 15136 df-pi 15138 df-dvds 15319 df-prm 15719 df-struct 16185 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-plusg 16279 df-mulr 16280 df-starv 16281 df-sca 16282 df-vsca 16283 df-ip 16284 df-tset 16285 df-ple 16286 df-ds 16288 df-unif 16289 df-hom 16290 df-cco 16291 df-rest 16397 df-topn 16398 df-0g 16416 df-gsum 16417 df-topgen 16418 df-pt 16419 df-prds 16422 df-xrs 16476 df-qtop 16481 df-imas 16482 df-xps 16484 df-mre 16560 df-mrc 16561 df-acs 16563 df-mgm 17556 df-sgrp 17598 df-mnd 17609 df-submnd 17650 df-mulg 17856 df-cntz 18061 df-cmn 18509 df-psmet 20059 df-xmet 20060 df-met 20061 df-bl 20062 df-mopn 20063 df-fbas 20064 df-fg 20065 df-cnfld 20068 df-top 21026 df-topon 21043 df-topsp 21065 df-bases 21078 df-cld 21151 df-ntr 21152 df-cls 21153 df-nei 21230 df-lp 21268 df-perf 21269 df-cn 21359 df-cnp 21360 df-haus 21447 df-tx 21693 df-hmeo 21886 df-fil 21977 df-fm 22069 df-flim 22070 df-flf 22071 df-xms 22452 df-ms 22453 df-tms 22454 df-cncf 23008 df-limc 23970 df-dv 23971 df-log 24643 df-cht 25174 |
This theorem is referenced by: cht3 25250 chtrpcl 25252 chtub 25288 |
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