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Mirrors > Home > ILE Home > Th. List > tanval2ap | Unicode version |
Description: Express the tangent
function directly in terms of ![]() |
Ref | Expression |
---|---|
tanval2ap |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tanvalap 11733 |
. . 3
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2 | 2cn 9007 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
3 | ax-icn 7923 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
4 | 2, 3 | mulcomi 7980 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 4 | oveq2i 5901 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | sinval 11727 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | adantr 276 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | simpl 109 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | mulcl 7955 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 3, 8, 9 | sylancr 414 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | efcl 11689 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 10, 11 | syl 14 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | negicn 8175 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() | |
14 | mulcl 7955 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 8, 14 | sylancr 414 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | efcl 11689 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | syl 14 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 12, 17 | subcld 8285 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 3 | a1i 9 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 2 | a1i 9 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | iap0 9159 |
. . . . . . 7
![]() ![]() ![]() | |
22 | 21 | a1i 9 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 2ap0 9029 |
. . . . . . 7
![]() ![]() ![]() | |
24 | 23 | a1i 9 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 18, 19, 20, 22, 24 | divdivap1d 8796 |
. . . . 5
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26 | 5, 7, 25 | 3eqtr4a 2247 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | cosval 11728 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 27 | adantr 276 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 26, 28 | oveq12d 5908 |
. . 3
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30 | 1, 29 | eqtrd 2221 |
. 2
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31 | 18, 19, 22 | divclapd 8764 |
. . 3
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32 | 12, 17 | addcld 7994 |
. . 3
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33 | simpr 110 |
. . . . 5
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34 | 28, 33 | eqbrtrrd 4041 |
. . . 4
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35 | 32, 20, 24 | divap0bd 8776 |
. . . 4
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36 | 34, 35 | mpbird 167 |
. . 3
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37 | 31, 32, 20, 36, 24 | divcanap7d 8793 |
. 2
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38 | 18, 19, 32, 22, 36 | divdivap1d 8796 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 30, 37, 38 | 3eqtrd 2225 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-nul 4143 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-iinf 4601 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-mulrcl 7927 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-precex 7938 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 ax-pre-mulgt0 7945 ax-pre-mulext 7946 ax-arch 7947 ax-caucvg 7948 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rmo 2475 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-if 3549 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-tr 4116 df-id 4307 df-po 4310 df-iso 4311 df-iord 4380 df-on 4382 df-ilim 4383 df-suc 4385 df-iom 4604 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-isom 5239 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-recs 6323 df-irdg 6388 df-frec 6409 df-1o 6434 df-oadd 6438 df-er 6552 df-en 6758 df-dom 6759 df-fin 6760 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-reap 8549 df-ap 8556 df-div 8647 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-n0 9194 df-z 9271 df-uz 9546 df-q 9637 df-rp 9671 df-ico 9911 df-fz 10026 df-fzo 10160 df-seqfrec 10463 df-exp 10537 df-fac 10723 df-ihash 10773 df-cj 10868 df-re 10869 df-im 10870 df-rsqrt 11024 df-abs 11025 df-clim 11304 df-sumdc 11379 df-ef 11673 df-sin 11675 df-cos 11676 df-tan 11677 |
This theorem is referenced by: tanval3ap 11739 |
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