Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sinadd | Unicode version |
Description: Addition formula for sine. Equation 14 of [Gleason] p. 310. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
sinadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcl 7745 | . . 3 | |
2 | sinval 11409 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | 2cn 8791 | . . . . . . 7 | |
5 | 4 | a1i 9 | . . . . . 6 |
6 | ax-icn 7715 | . . . . . . 7 | |
7 | 6 | a1i 9 | . . . . . 6 |
8 | coscl 11414 | . . . . . . . . 9 | |
9 | 8 | adantr 274 | . . . . . . . 8 |
10 | sincl 11413 | . . . . . . . . 9 | |
11 | 10 | adantl 275 | . . . . . . . 8 |
12 | 9, 11 | mulcld 7786 | . . . . . . 7 |
13 | sincl 11413 | . . . . . . . . 9 | |
14 | 13 | adantr 274 | . . . . . . . 8 |
15 | coscl 11414 | . . . . . . . . 9 | |
16 | 15 | adantl 275 | . . . . . . . 8 |
17 | 14, 16 | mulcld 7786 | . . . . . . 7 |
18 | 12, 17 | addcld 7785 | . . . . . 6 |
19 | 5, 7, 18 | mulassd 7789 | . . . . 5 |
20 | 7, 12, 17 | adddid 7790 | . . . . . . 7 |
21 | 7, 9, 11 | mul12d 7914 | . . . . . . . 8 |
22 | 14, 16 | mulcomd 7787 | . . . . . . . . . 10 |
23 | 22 | oveq2d 5790 | . . . . . . . . 9 |
24 | 7, 16, 14 | mul12d 7914 | . . . . . . . . 9 |
25 | 23, 24 | eqtrd 2172 | . . . . . . . 8 |
26 | 21, 25 | oveq12d 5792 | . . . . . . 7 |
27 | 20, 26 | eqtrd 2172 | . . . . . 6 |
28 | 27 | oveq2d 5790 | . . . . 5 |
29 | 19, 28 | eqtrd 2172 | . . . 4 |
30 | mulcl 7747 | . . . . . . . . 9 | |
31 | 6, 11, 30 | sylancr 410 | . . . . . . . 8 |
32 | 9, 31 | mulcld 7786 | . . . . . . 7 |
33 | mulcl 7747 | . . . . . . . . 9 | |
34 | 6, 14, 33 | sylancr 410 | . . . . . . . 8 |
35 | 16, 34 | mulcld 7786 | . . . . . . 7 |
36 | 32, 35 | addcld 7785 | . . . . . 6 |
37 | mulcl 7747 | . . . . . 6 | |
38 | 4, 36, 37 | sylancr 410 | . . . . 5 |
39 | 2mulicn 8942 | . . . . . 6 | |
40 | 39 | a1i 9 | . . . . 5 |
41 | 2muliap0 8944 | . . . . . 6 # | |
42 | 41 | a1i 9 | . . . . 5 # |
43 | 38, 40, 18, 42 | divmulapd 8572 | . . . 4 |
44 | 29, 43 | mpbird 166 | . . 3 |
45 | 9, 16 | mulcld 7786 | . . . . . . 7 |
46 | 31, 34 | mulcld 7786 | . . . . . . 7 |
47 | 45, 46 | addcld 7785 | . . . . . 6 |
48 | 47, 36, 36 | pnncand 8112 | . . . . 5 |
49 | adddi 7752 | . . . . . . . . 9 | |
50 | 6, 49 | mp3an1 1302 | . . . . . . . 8 |
51 | 50 | fveq2d 5425 | . . . . . . 7 |
52 | simpl 108 | . . . . . . . . 9 | |
53 | mulcl 7747 | . . . . . . . . 9 | |
54 | 6, 52, 53 | sylancr 410 | . . . . . . . 8 |
55 | simpr 109 | . . . . . . . . 9 | |
56 | mulcl 7747 | . . . . . . . . 9 | |
57 | 6, 55, 56 | sylancr 410 | . . . . . . . 8 |
58 | efadd 11381 | . . . . . . . 8 | |
59 | 54, 57, 58 | syl2anc 408 | . . . . . . 7 |
60 | efival 11439 | . . . . . . . . 9 | |
61 | efival 11439 | . . . . . . . . 9 | |
62 | 60, 61 | oveqan12d 5793 | . . . . . . . 8 |
63 | 9, 34, 16, 31 | muladdd 8178 | . . . . . . . 8 |
64 | 62, 63 | eqtrd 2172 | . . . . . . 7 |
65 | 51, 59, 64 | 3eqtrd 2176 | . . . . . 6 |
66 | negicn 7963 | . . . . . . . . 9 | |
67 | adddi 7752 | . . . . . . . . 9 | |
68 | 66, 67 | mp3an1 1302 | . . . . . . . 8 |
69 | 68 | fveq2d 5425 | . . . . . . 7 |
70 | mulcl 7747 | . . . . . . . . 9 | |
71 | 66, 52, 70 | sylancr 410 | . . . . . . . 8 |
72 | mulcl 7747 | . . . . . . . . 9 | |
73 | 66, 55, 72 | sylancr 410 | . . . . . . . 8 |
74 | efadd 11381 | . . . . . . . 8 | |
75 | 71, 73, 74 | syl2anc 408 | . . . . . . 7 |
76 | efmival 11440 | . . . . . . . . 9 | |
77 | efmival 11440 | . . . . . . . . 9 | |
78 | 76, 77 | oveqan12d 5793 | . . . . . . . 8 |
79 | 9, 34, 16, 31 | mulsubd 8179 | . . . . . . . 8 |
80 | 78, 79 | eqtrd 2172 | . . . . . . 7 |
81 | 69, 75, 80 | 3eqtrd 2176 | . . . . . 6 |
82 | 65, 81 | oveq12d 5792 | . . . . 5 |
83 | 36 | 2timesd 8962 | . . . . 5 |
84 | 48, 82, 83 | 3eqtr4d 2182 | . . . 4 |
85 | 84 | oveq1d 5789 | . . 3 |
86 | 17, 12 | addcomd 7913 | . . 3 |
87 | 44, 85, 86 | 3eqtr4d 2182 | . 2 |
88 | 3, 87 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7618 cc0 7620 ci 7622 caddc 7623 cmul 7625 cmin 7933 cneg 7934 # cap 8343 cdiv 8432 c2 8771 ce 11348 csin 11350 ccos 11351 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-disj 3907 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-bc 10494 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 df-ef 11354 df-sin 11356 df-cos 11357 |
This theorem is referenced by: tanaddap 11446 sinsub 11447 addsin 11449 subsin 11450 sin2t 11456 demoivreALT 11480 sinppi 12898 sinhalfpip 12901 |
Copyright terms: Public domain | W3C validator |