![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sinmpi | Structured version Visualization version GIF version |
Description: Sine of a number less π. (Contributed by Paul Chapman, 15-Mar-2008.) |
Ref | Expression |
---|---|
sinmpi | ⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | picn 26521 | . . 3 ⊢ π ∈ ℂ | |
2 | sinsub 16218 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ π ∈ ℂ) → (sin‘(𝐴 − π)) = (((sin‘𝐴) · (cos‘π)) − ((cos‘𝐴) · (sin‘π)))) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = (((sin‘𝐴) · (cos‘π)) − ((cos‘𝐴) · (sin‘π)))) |
4 | cospi 26534 | . . . . . 6 ⊢ (cos‘π) = -1 | |
5 | 4 | oveq2i 7461 | . . . . 5 ⊢ ((sin‘𝐴) · (cos‘π)) = ((sin‘𝐴) · -1) |
6 | sincl 16176 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
7 | neg1cn 12409 | . . . . . . . 8 ⊢ -1 ∈ ℂ | |
8 | mulcom 11272 | . . . . . . . 8 ⊢ (((sin‘𝐴) ∈ ℂ ∧ -1 ∈ ℂ) → ((sin‘𝐴) · -1) = (-1 · (sin‘𝐴))) | |
9 | 7, 8 | mpan2 690 | . . . . . . 7 ⊢ ((sin‘𝐴) ∈ ℂ → ((sin‘𝐴) · -1) = (-1 · (sin‘𝐴))) |
10 | mulm1 11733 | . . . . . . 7 ⊢ ((sin‘𝐴) ∈ ℂ → (-1 · (sin‘𝐴)) = -(sin‘𝐴)) | |
11 | 9, 10 | eqtrd 2780 | . . . . . 6 ⊢ ((sin‘𝐴) ∈ ℂ → ((sin‘𝐴) · -1) = -(sin‘𝐴)) |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -1) = -(sin‘𝐴)) |
13 | 5, 12 | eqtrid 2792 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · (cos‘π)) = -(sin‘𝐴)) |
14 | sinpi 26519 | . . . . . 6 ⊢ (sin‘π) = 0 | |
15 | 14 | oveq2i 7461 | . . . . 5 ⊢ ((cos‘𝐴) · (sin‘π)) = ((cos‘𝐴) · 0) |
16 | coscl 16177 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
17 | 16 | mul01d 11491 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · 0) = 0) |
18 | 15, 17 | eqtrid 2792 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (sin‘π)) = 0) |
19 | 13, 18 | oveq12d 7468 | . . 3 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴) · (cos‘π)) − ((cos‘𝐴) · (sin‘π))) = (-(sin‘𝐴) − 0)) |
20 | 6 | negcld 11636 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(sin‘𝐴) ∈ ℂ) |
21 | 20 | subid1d 11638 | . . 3 ⊢ (𝐴 ∈ ℂ → (-(sin‘𝐴) − 0) = -(sin‘𝐴)) |
22 | 19, 21 | eqtrd 2780 | . 2 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴) · (cos‘π)) − ((cos‘𝐴) · (sin‘π))) = -(sin‘𝐴)) |
23 | 3, 22 | eqtrd 2780 | 1 ⊢ (𝐴 ∈ ℂ → (sin‘(𝐴 − π)) = -(sin‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ‘cfv 6575 (class class class)co 7450 ℂcc 11184 0cc0 11186 1c1 11187 · cmul 11191 − cmin 11522 -cneg 11523 sincsin 16113 cosccos 16114 πcpi 16116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 ax-addf 11265 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-supp 8204 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-map 8888 df-pm 8889 df-ixp 8958 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-fsupp 9434 df-fi 9482 df-sup 9513 df-inf 9514 df-oi 9581 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-q 13016 df-rp 13060 df-xneg 13177 df-xadd 13178 df-xmul 13179 df-ioo 13413 df-ioc 13414 df-ico 13415 df-icc 13416 df-fz 13570 df-fzo 13714 df-fl 13845 df-seq 14055 df-exp 14115 df-fac 14325 df-bc 14354 df-hash 14382 df-shft 15118 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-limsup 15519 df-clim 15536 df-rlim 15537 df-sum 15737 df-ef 16117 df-sin 16119 df-cos 16120 df-pi 16122 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-rest 17484 df-topn 17485 df-0g 17503 df-gsum 17504 df-topgen 17505 df-pt 17506 df-prds 17509 df-xrs 17564 df-qtop 17569 df-imas 17570 df-xps 17572 df-mre 17646 df-mrc 17647 df-acs 17649 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-mulg 19110 df-cntz 19359 df-cmn 19826 df-psmet 21381 df-xmet 21382 df-met 21383 df-bl 21384 df-mopn 21385 df-fbas 21386 df-fg 21387 df-cnfld 21390 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22976 df-cld 23050 df-ntr 23051 df-cls 23052 df-nei 23129 df-lp 23167 df-perf 23168 df-cn 23258 df-cnp 23259 df-haus 23346 df-tx 23593 df-hmeo 23786 df-fil 23877 df-fm 23969 df-flim 23970 df-flf 23971 df-xms 24353 df-ms 24354 df-tms 24355 df-cncf 24925 df-limc 25923 df-dv 25924 |
This theorem is referenced by: efimpi 26553 sinq34lt0t 26571 abssinper 26583 |
Copyright terms: Public domain | W3C validator |