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Mirrors > Home > MPE Home > Th. List > cosppi | Structured version Visualization version GIF version |
Description: Cosine of a number plus π. (Contributed by NM, 18-Aug-2008.) |
Ref | Expression |
---|---|
cosppi | ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | picn 25043 | . . 3 ⊢ π ∈ ℂ | |
2 | cosadd 15514 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ π ∈ ℂ) → (cos‘(𝐴 + π)) = (((cos‘𝐴) · (cos‘π)) − ((sin‘𝐴) · (sin‘π)))) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = (((cos‘𝐴) · (cos‘π)) − ((sin‘𝐴) · (sin‘π)))) |
4 | cospi 25056 | . . . . . 6 ⊢ (cos‘π) = -1 | |
5 | 4 | oveq2i 7164 | . . . . 5 ⊢ ((cos‘𝐴) · (cos‘π)) = ((cos‘𝐴) · -1) |
6 | coscl 15476 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
7 | neg1cn 11749 | . . . . . . . 8 ⊢ -1 ∈ ℂ | |
8 | mulcom 10620 | . . . . . . . 8 ⊢ (((cos‘𝐴) ∈ ℂ ∧ -1 ∈ ℂ) → ((cos‘𝐴) · -1) = (-1 · (cos‘𝐴))) | |
9 | 7, 8 | mpan2 689 | . . . . . . 7 ⊢ ((cos‘𝐴) ∈ ℂ → ((cos‘𝐴) · -1) = (-1 · (cos‘𝐴))) |
10 | mulm1 11078 | . . . . . . 7 ⊢ ((cos‘𝐴) ∈ ℂ → (-1 · (cos‘𝐴)) = -(cos‘𝐴)) | |
11 | 9, 10 | eqtrd 2855 | . . . . . 6 ⊢ ((cos‘𝐴) ∈ ℂ → ((cos‘𝐴) · -1) = -(cos‘𝐴)) |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · -1) = -(cos‘𝐴)) |
13 | 5, 12 | syl5eq 2867 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘π)) = -(cos‘𝐴)) |
14 | sinpi 25041 | . . . . . 6 ⊢ (sin‘π) = 0 | |
15 | 14 | oveq2i 7164 | . . . . 5 ⊢ ((sin‘𝐴) · (sin‘π)) = ((sin‘𝐴) · 0) |
16 | sincl 15475 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
17 | 16 | mul01d 10836 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · 0) = 0) |
18 | 15, 17 | syl5eq 2867 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · (sin‘π)) = 0) |
19 | 13, 18 | oveq12d 7171 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘π)) − ((sin‘𝐴) · (sin‘π))) = (-(cos‘𝐴) − 0)) |
20 | 6 | negcld 10981 | . . . 4 ⊢ (𝐴 ∈ ℂ → -(cos‘𝐴) ∈ ℂ) |
21 | 20 | subid1d 10983 | . . 3 ⊢ (𝐴 ∈ ℂ → (-(cos‘𝐴) − 0) = -(cos‘𝐴)) |
22 | 19, 21 | eqtrd 2855 | . 2 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘π)) − ((sin‘𝐴) · (sin‘π))) = -(cos‘𝐴)) |
23 | 3, 22 | eqtrd 2855 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + π)) = -(cos‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6352 (class class class)co 7153 ℂcc 10532 0cc0 10534 1c1 10535 + caddc 10537 · cmul 10539 − cmin 10867 -cneg 10868 sincsin 15413 cosccos 15414 πcpi 15416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-inf2 9101 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 ax-pre-sup 10612 ax-addf 10613 ax-mulf 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-iin 4919 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-se 5512 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-isom 6361 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-of 7406 df-om 7578 df-1st 7686 df-2nd 7687 df-supp 7828 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-oadd 8103 df-er 8286 df-map 8405 df-pm 8406 df-ixp 8459 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-fsupp 8831 df-fi 8872 df-sup 8903 df-inf 8904 df-oi 8971 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-div 11295 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-z 11980 df-dec 12097 df-uz 12242 df-q 12347 df-rp 12388 df-xneg 12505 df-xadd 12506 df-xmul 12507 df-ioo 12740 df-ioc 12741 df-ico 12742 df-icc 12743 df-fz 12891 df-fzo 13032 df-fl 13160 df-seq 13368 df-exp 13428 df-fac 13632 df-bc 13661 df-hash 13689 df-shft 14422 df-cj 14454 df-re 14455 df-im 14456 df-sqrt 14590 df-abs 14591 df-limsup 14824 df-clim 14841 df-rlim 14842 df-sum 15039 df-ef 15417 df-sin 15419 df-cos 15420 df-pi 15422 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-starv 16576 df-sca 16577 df-vsca 16578 df-ip 16579 df-tset 16580 df-ple 16581 df-ds 16583 df-unif 16584 df-hom 16585 df-cco 16586 df-rest 16692 df-topn 16693 df-0g 16711 df-gsum 16712 df-topgen 16713 df-pt 16714 df-prds 16717 df-xrs 16771 df-qtop 16776 df-imas 16777 df-xps 16779 df-mre 16853 df-mrc 16854 df-acs 16856 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-submnd 17953 df-mulg 18221 df-cntz 18443 df-cmn 18904 df-psmet 20533 df-xmet 20534 df-met 20535 df-bl 20536 df-mopn 20537 df-fbas 20538 df-fg 20539 df-cnfld 20542 df-top 21498 df-topon 21515 df-topsp 21537 df-bases 21550 df-cld 21623 df-ntr 21624 df-cls 21625 df-nei 21702 df-lp 21740 df-perf 21741 df-cn 21831 df-cnp 21832 df-haus 21919 df-tx 22166 df-hmeo 22359 df-fil 22450 df-fm 22542 df-flim 22543 df-flf 22544 df-xms 22926 df-ms 22927 df-tms 22928 df-cncf 23482 df-limc 24462 df-dv 24463 |
This theorem is referenced by: ptolemy 25080 |
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