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Mirrors > Home > ILE Home > Th. List > cosadd | Unicode version |
Description: Addition formula for cosine. Equation 15 of [Gleason] p. 310. (Contributed by NM, 15-Jan-2006.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
cosadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcl 7899 | . . 3 | |
2 | cosval 11666 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | coscl 11670 | . . . . . . . 8 | |
5 | 4 | adantr 274 | . . . . . . 7 |
6 | coscl 11670 | . . . . . . . 8 | |
7 | 6 | adantl 275 | . . . . . . 7 |
8 | 5, 7 | mulcld 7940 | . . . . . 6 |
9 | ax-icn 7869 | . . . . . . . 8 | |
10 | sincl 11669 | . . . . . . . . 9 | |
11 | 10 | adantl 275 | . . . . . . . 8 |
12 | mulcl 7901 | . . . . . . . 8 | |
13 | 9, 11, 12 | sylancr 412 | . . . . . . 7 |
14 | sincl 11669 | . . . . . . . . 9 | |
15 | 14 | adantr 274 | . . . . . . . 8 |
16 | mulcl 7901 | . . . . . . . 8 | |
17 | 9, 15, 16 | sylancr 412 | . . . . . . 7 |
18 | 13, 17 | mulcld 7940 | . . . . . 6 |
19 | 8, 18 | addcld 7939 | . . . . 5 |
20 | 5, 13 | mulcld 7940 | . . . . . 6 |
21 | 7, 17 | mulcld 7940 | . . . . . 6 |
22 | 20, 21 | addcld 7939 | . . . . 5 |
23 | 19, 22, 19 | ppncand 8270 | . . . 4 |
24 | adddi 7906 | . . . . . . . 8 | |
25 | 9, 24 | mp3an1 1319 | . . . . . . 7 |
26 | 25 | fveq2d 5500 | . . . . . 6 |
27 | simpl 108 | . . . . . . . 8 | |
28 | mulcl 7901 | . . . . . . . 8 | |
29 | 9, 27, 28 | sylancr 412 | . . . . . . 7 |
30 | simpr 109 | . . . . . . . 8 | |
31 | mulcl 7901 | . . . . . . . 8 | |
32 | 9, 30, 31 | sylancr 412 | . . . . . . 7 |
33 | efadd 11638 | . . . . . . 7 | |
34 | 29, 32, 33 | syl2anc 409 | . . . . . 6 |
35 | efival 11695 | . . . . . . . 8 | |
36 | efival 11695 | . . . . . . . 8 | |
37 | 35, 36 | oveqan12d 5872 | . . . . . . 7 |
38 | 5, 17, 7, 13 | muladdd 8335 | . . . . . . 7 |
39 | 37, 38 | eqtrd 2203 | . . . . . 6 |
40 | 26, 34, 39 | 3eqtrd 2207 | . . . . 5 |
41 | negicn 8120 | . . . . . . . 8 | |
42 | adddi 7906 | . . . . . . . 8 | |
43 | 41, 42 | mp3an1 1319 | . . . . . . 7 |
44 | 43 | fveq2d 5500 | . . . . . 6 |
45 | mulcl 7901 | . . . . . . . 8 | |
46 | 41, 27, 45 | sylancr 412 | . . . . . . 7 |
47 | mulcl 7901 | . . . . . . . 8 | |
48 | 41, 30, 47 | sylancr 412 | . . . . . . 7 |
49 | efadd 11638 | . . . . . . 7 | |
50 | 46, 48, 49 | syl2anc 409 | . . . . . 6 |
51 | efmival 11696 | . . . . . . . 8 | |
52 | efmival 11696 | . . . . . . . 8 | |
53 | 51, 52 | oveqan12d 5872 | . . . . . . 7 |
54 | 5, 17, 7, 13 | mulsubd 8336 | . . . . . . 7 |
55 | 53, 54 | eqtrd 2203 | . . . . . 6 |
56 | 44, 50, 55 | 3eqtrd 2207 | . . . . 5 |
57 | 40, 56 | oveq12d 5871 | . . . 4 |
58 | 19 | 2timesd 9120 | . . . 4 |
59 | 23, 57, 58 | 3eqtr4d 2213 | . . 3 |
60 | 59 | oveq1d 5868 | . 2 |
61 | 2cn 8949 | . . . . 5 | |
62 | 2ap0 8971 | . . . . 5 # | |
63 | divcanap3 8615 | . . . . 5 # | |
64 | 61, 62, 63 | mp3an23 1324 | . . . 4 |
65 | 19, 64 | syl 14 | . . 3 |
66 | 9 | a1i 9 | . . . . . 6 |
67 | 66, 11, 66, 15 | mul4d 8074 | . . . . 5 |
68 | ixi 8502 | . . . . . . 7 | |
69 | 68 | oveq1i 5863 | . . . . . 6 |
70 | 11, 15 | mulcomd 7941 | . . . . . . 7 |
71 | 70 | oveq2d 5869 | . . . . . 6 |
72 | 69, 71 | eqtrid 2215 | . . . . 5 |
73 | 15, 11 | mulcld 7940 | . . . . . 6 |
74 | 73 | mulm1d 8329 | . . . . 5 |
75 | 67, 72, 74 | 3eqtrd 2207 | . . . 4 |
76 | 75 | oveq2d 5869 | . . 3 |
77 | 8, 73 | negsubd 8236 | . . 3 |
78 | 65, 76, 77 | 3eqtrd 2207 | . 2 |
79 | 3, 60, 78 | 3eqtrd 2207 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 class class class wbr 3989 cfv 5198 (class class class)co 5853 cc 7772 cc0 7774 c1 7775 ci 7776 caddc 7777 cmul 7779 cmin 8090 cneg 8091 # cap 8500 cdiv 8589 c2 8929 ce 11605 csin 11607 ccos 11608 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-disj 3967 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-ico 9851 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-fac 10660 df-bc 10682 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 df-ef 11611 df-sin 11613 df-cos 11614 |
This theorem is referenced by: tanaddaplem 11701 tanaddap 11702 cossub 11704 sinmul 11707 cosmul 11708 addcos 11709 subcos 11710 sincossq 11711 cos2t 11713 cos12dec 11730 demoivreALT 11736 cosppi 13533 coshalfpip 13537 |
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