Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cht3 | Structured version Visualization version GIF version |
Description: The Chebyshev function at 3. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
cht3 | ⊢ (θ‘3) = (log‘6) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3 11690 | . . 3 ⊢ 3 = (2 + 1) | |
2 | 1 | fveq2i 6667 | . 2 ⊢ (θ‘3) = (θ‘(2 + 1)) |
3 | 2z 12003 | . . 3 ⊢ 2 ∈ ℤ | |
4 | 3prm 16028 | . . . 4 ⊢ 3 ∈ ℙ | |
5 | 1, 4 | eqeltrri 2910 | . . 3 ⊢ (2 + 1) ∈ ℙ |
6 | chtprm 25658 | . . 3 ⊢ ((2 ∈ ℤ ∧ (2 + 1) ∈ ℙ) → (θ‘(2 + 1)) = ((θ‘2) + (log‘(2 + 1)))) | |
7 | 3, 5, 6 | mp2an 688 | . 2 ⊢ (θ‘(2 + 1)) = ((θ‘2) + (log‘(2 + 1))) |
8 | 2rp 12384 | . . . 4 ⊢ 2 ∈ ℝ+ | |
9 | 3rp 12385 | . . . 4 ⊢ 3 ∈ ℝ+ | |
10 | relogmul 25102 | . . . 4 ⊢ ((2 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (log‘(2 · 3)) = ((log‘2) + (log‘3))) | |
11 | 8, 9, 10 | mp2an 688 | . . 3 ⊢ (log‘(2 · 3)) = ((log‘2) + (log‘3)) |
12 | 3cn 11707 | . . . . 5 ⊢ 3 ∈ ℂ | |
13 | 2cn 11701 | . . . . 5 ⊢ 2 ∈ ℂ | |
14 | 3t2e6 11792 | . . . . 5 ⊢ (3 · 2) = 6 | |
15 | 12, 13, 14 | mulcomli 10639 | . . . 4 ⊢ (2 · 3) = 6 |
16 | 15 | fveq2i 6667 | . . 3 ⊢ (log‘(2 · 3)) = (log‘6) |
17 | cht2 25677 | . . . . 5 ⊢ (θ‘2) = (log‘2) | |
18 | 17 | eqcomi 2830 | . . . 4 ⊢ (log‘2) = (θ‘2) |
19 | 1 | fveq2i 6667 | . . . 4 ⊢ (log‘3) = (log‘(2 + 1)) |
20 | 18, 19 | oveq12i 7157 | . . 3 ⊢ ((log‘2) + (log‘3)) = ((θ‘2) + (log‘(2 + 1))) |
21 | 11, 16, 20 | 3eqtr3ri 2853 | . 2 ⊢ ((θ‘2) + (log‘(2 + 1))) = (log‘6) |
22 | 2, 7, 21 | 3eqtri 2848 | 1 ⊢ (θ‘3) = (log‘6) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 1c1 10527 + caddc 10529 · cmul 10531 2c2 11681 3c3 11682 6c6 11685 ℤcz 11970 ℝ+crp 12379 ℙcprime 16005 logclog 25065 θccht 25596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-fi 8864 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-ioo 12732 df-ioc 12733 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14416 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-limsup 14818 df-clim 14835 df-rlim 14836 df-sum 15033 df-ef 15411 df-sin 15413 df-cos 15414 df-pi 15416 df-dvds 15598 df-prm 16006 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-rest 16686 df-topn 16687 df-0g 16705 df-gsum 16706 df-topgen 16707 df-pt 16708 df-prds 16711 df-xrs 16765 df-qtop 16770 df-imas 16771 df-xps 16773 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-submnd 17947 df-mulg 18165 df-cntz 18387 df-cmn 18839 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-fbas 20472 df-fg 20473 df-cnfld 20476 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cld 21557 df-ntr 21558 df-cls 21559 df-nei 21636 df-lp 21674 df-perf 21675 df-cn 21765 df-cnp 21766 df-haus 21853 df-tx 22100 df-hmeo 22293 df-fil 22384 df-fm 22476 df-flim 22477 df-flf 22478 df-xms 22859 df-ms 22860 df-tms 22861 df-cncf 23415 df-limc 24393 df-dv 24394 df-log 25067 df-cht 25602 |
This theorem is referenced by: chtub 25716 bposlem6 25793 |
Copyright terms: Public domain | W3C validator |