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Mirrors > Home > MPE Home > Th. List > tan4thpi | Structured version Unicode version |
Description: The tangent of ![]() ![]() ![]() |
Ref | Expression |
---|---|
tan4thpi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pire 22053 |
. . . . 5
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2 | 4nn 10591 |
. . . . 5
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3 | nndivre 10467 |
. . . . 5
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4 | 1, 2, 3 | mp2an 672 |
. . . 4
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5 | 4 | recni 9508 |
. . 3
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6 | sincos4thpi 22107 |
. . . . 5
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7 | 6 | simpri 462 |
. . . 4
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8 | sqr2re 13649 |
. . . . . 6
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9 | 8 | recni 9508 |
. . . . 5
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10 | 2re 10501 |
. . . . . . . 8
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11 | 0le2 10522 |
. . . . . . . 8
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12 | resqrth 12862 |
. . . . . . . 8
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13 | 10, 11, 12 | mp2an 672 |
. . . . . . 7
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14 | 2ne0 10524 |
. . . . . . 7
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15 | 13, 14 | eqnetri 2747 |
. . . . . 6
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16 | sqne0 12048 |
. . . . . . 7
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17 | 9, 16 | ax-mp 5 |
. . . . . 6
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18 | 15, 17 | mpbi 208 |
. . . . 5
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19 | recne0 10117 |
. . . . 5
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20 | 9, 18, 19 | mp2an 672 |
. . . 4
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21 | 7, 20 | eqnetri 2747 |
. . 3
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22 | tanval 13529 |
. . 3
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23 | 5, 21, 22 | mp2an 672 |
. 2
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24 | 6 | simpli 458 |
. . 3
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25 | 24, 7 | oveq12i 6211 |
. 2
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26 | 9, 18 | reccli 10171 |
. . 3
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27 | 26, 20 | dividi 10174 |
. 2
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28 | 23, 25, 27 | 3eqtri 2487 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4510 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 ax-inf2 7957 ax-cnex 9448 ax-resscn 9449 ax-1cn 9450 ax-icn 9451 ax-addcl 9452 ax-addrcl 9453 ax-mulcl 9454 ax-mulrcl 9455 ax-mulcom 9456 ax-addass 9457 ax-mulass 9458 ax-distr 9459 ax-i2m1 9460 ax-1ne0 9461 ax-1rid 9462 ax-rnegex 9463 ax-rrecex 9464 ax-cnre 9465 ax-pre-lttri 9466 ax-pre-lttrn 9467 ax-pre-ltadd 9468 ax-pre-mulgt0 9469 ax-pre-sup 9470 ax-addf 9471 ax-mulf 9472 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-nel 2650 df-ral 2803 df-rex 2804 df-reu 2805 df-rmo 2806 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-pss 3451 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-tp 3989 df-op 3991 df-uni 4199 df-int 4236 df-iun 4280 df-iin 4281 df-br 4400 df-opab 4458 df-mpt 4459 df-tr 4493 df-eprel 4739 df-id 4743 df-po 4748 df-so 4749 df-fr 4786 df-se 4787 df-we 4788 df-ord 4829 df-on 4830 df-lim 4831 df-suc 4832 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-isom 5534 df-riota 6160 df-ov 6202 df-oprab 6203 df-mpt2 6204 df-of 6429 df-om 6586 df-1st 6686 df-2nd 6687 df-supp 6800 df-recs 6941 df-rdg 6975 df-1o 7029 df-2o 7030 df-oadd 7033 df-er 7210 df-map 7325 df-pm 7326 df-ixp 7373 df-en 7420 df-dom 7421 df-sdom 7422 df-fin 7423 df-fsupp 7731 df-fi 7771 df-sup 7801 df-oi 7834 df-card 8219 df-cda 8447 df-pnf 9530 df-mnf 9531 df-xr 9532 df-ltxr 9533 df-le 9534 df-sub 9707 df-neg 9708 df-div 10104 df-nn 10433 df-2 10490 df-3 10491 df-4 10492 df-5 10493 df-6 10494 df-7 10495 df-8 10496 df-9 10497 df-10 10498 df-n0 10690 df-z 10757 df-dec 10866 df-uz 10972 df-q 11064 df-rp 11102 df-xneg 11199 df-xadd 11200 df-xmul 11201 df-ioo 11414 df-ioc 11415 df-ico 11416 df-icc 11417 df-fz 11554 df-fzo 11665 df-fl 11758 df-seq 11923 df-exp 11982 df-fac 12168 df-bc 12195 df-hash 12220 df-shft 12673 df-cj 12705 df-re 12706 df-im 12707 df-sqr 12841 df-abs 12842 df-limsup 13066 df-clim 13083 df-rlim 13084 df-sum 13281 df-ef 13470 df-sin 13472 df-cos 13473 df-tan 13474 df-pi 13475 df-struct 14293 df-ndx 14294 df-slot 14295 df-base 14296 df-sets 14297 df-ress 14298 df-plusg 14369 df-mulr 14370 df-starv 14371 df-sca 14372 df-vsca 14373 df-ip 14374 df-tset 14375 df-ple 14376 df-ds 14378 df-unif 14379 df-hom 14380 df-cco 14381 df-rest 14479 df-topn 14480 df-0g 14498 df-gsum 14499 df-topgen 14500 df-pt 14501 df-prds 14504 df-xrs 14558 df-qtop 14563 df-imas 14564 df-xps 14566 df-mre 14642 df-mrc 14643 df-acs 14645 df-mnd 15533 df-submnd 15583 df-mulg 15666 df-cntz 15953 df-cmn 16399 df-psmet 17933 df-xmet 17934 df-met 17935 df-bl 17936 df-mopn 17937 df-fbas 17938 df-fg 17939 df-cnfld 17943 df-top 18634 df-bases 18636 df-topon 18637 df-topsp 18638 df-cld 18754 df-ntr 18755 df-cls 18756 df-nei 18833 df-lp 18871 df-perf 18872 df-cn 18962 df-cnp 18963 df-haus 19050 df-tx 19266 df-hmeo 19459 df-fil 19550 df-fm 19642 df-flim 19643 df-flf 19644 df-xms 20026 df-ms 20027 df-tms 20028 df-cncf 20585 df-limc 21473 df-dv 21474 |
This theorem is referenced by: atan1 22455 |
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