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Mirrors > Home > ILE Home > Th. List > sin2t | Unicode version |
Description: Double-angle formula for sine. (Contributed by Paul Chapman, 17-Jan-2008.) |
Ref | Expression |
---|---|
sin2t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2times 8981 | . . 3 | |
2 | 1 | fveq2d 5489 | . 2 |
3 | coscl 11644 | . . . . 5 | |
4 | sincl 11643 | . . . . 5 | |
5 | 3, 4 | mulcomd 7916 | . . . 4 |
6 | 5 | oveq2d 5857 | . . 3 |
7 | sinadd 11673 | . . . 4 | |
8 | 7 | anidms 395 | . . 3 |
9 | 4, 3 | mulcld 7915 | . . . 4 |
10 | 9 | 2timesd 9095 | . . 3 |
11 | 6, 8, 10 | 3eqtr4d 2208 | . 2 |
12 | 2, 11 | eqtrd 2198 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1343 wcel 2136 cfv 5187 (class class class)co 5841 cc 7747 caddc 7752 cmul 7754 c2 8904 csin 11581 ccos 11582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4096 ax-sep 4099 ax-nul 4107 ax-pow 4152 ax-pr 4186 ax-un 4410 ax-setind 4513 ax-iinf 4564 ax-cnex 7840 ax-resscn 7841 ax-1cn 7842 ax-1re 7843 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-mulrcl 7848 ax-addcom 7849 ax-mulcom 7850 ax-addass 7851 ax-mulass 7852 ax-distr 7853 ax-i2m1 7854 ax-0lt1 7855 ax-1rid 7856 ax-0id 7857 ax-rnegex 7858 ax-precex 7859 ax-cnre 7860 ax-pre-ltirr 7861 ax-pre-ltwlin 7862 ax-pre-lttrn 7863 ax-pre-apti 7864 ax-pre-ltadd 7865 ax-pre-mulgt0 7866 ax-pre-mulext 7867 ax-arch 7868 ax-caucvg 7869 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ne 2336 df-nel 2431 df-ral 2448 df-rex 2449 df-reu 2450 df-rmo 2451 df-rab 2452 df-v 2727 df-sbc 2951 df-csb 3045 df-dif 3117 df-un 3119 df-in 3121 df-ss 3128 df-nul 3409 df-if 3520 df-pw 3560 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-int 3824 df-iun 3867 df-disj 3959 df-br 3982 df-opab 4043 df-mpt 4044 df-tr 4080 df-id 4270 df-po 4273 df-iso 4274 df-iord 4343 df-on 4345 df-ilim 4346 df-suc 4348 df-iom 4567 df-xp 4609 df-rel 4610 df-cnv 4611 df-co 4612 df-dm 4613 df-rn 4614 df-res 4615 df-ima 4616 df-iota 5152 df-fun 5189 df-fn 5190 df-f 5191 df-f1 5192 df-fo 5193 df-f1o 5194 df-fv 5195 df-isom 5196 df-riota 5797 df-ov 5844 df-oprab 5845 df-mpo 5846 df-1st 6105 df-2nd 6106 df-recs 6269 df-irdg 6334 df-frec 6355 df-1o 6380 df-oadd 6384 df-er 6497 df-en 6703 df-dom 6704 df-fin 6705 df-sup 6945 df-pnf 7931 df-mnf 7932 df-xr 7933 df-ltxr 7934 df-le 7935 df-sub 8067 df-neg 8068 df-reap 8469 df-ap 8476 df-div 8565 df-inn 8854 df-2 8912 df-3 8913 df-4 8914 df-n0 9111 df-z 9188 df-uz 9463 df-q 9554 df-rp 9586 df-ico 9826 df-fz 9941 df-fzo 10074 df-seqfrec 10377 df-exp 10451 df-fac 10635 df-bc 10657 df-ihash 10685 df-cj 10780 df-re 10781 df-im 10782 df-rsqrt 10936 df-abs 10937 df-clim 11216 df-sumdc 11291 df-ef 11585 df-sin 11587 df-cos 11588 |
This theorem is referenced by: sin02gt0 11700 sin4lt0 11703 sin0pilem1 13302 sin0pilem2 13303 sinhalfpilem 13312 sin2pi 13324 sinq12gt0 13351 tangtx 13359 sincos4thpi 13361 sincos6thpi 13363 |
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