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| Mirrors > Home > ILE Home > Th. List > tanaddap | Unicode version | ||
| Description: Addition formula for tangent. (Contributed by Mario Carneiro, 4-Apr-2015.) |
| Ref | Expression |
|---|---|
| tanaddap |
     ![/\](wedge.gif "/\"){kind=small}     #  #  #        |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | addcl 8124 |
. . . 4
| |
| 2 | 1 | adantr 276 |
. . 3
# # #
|
| 3 | simpr3 1029 |
. . 3
# # # # | |
| 4 | tanvalap 12219 |
. . 3
# 
| |
| 5 | 2, 3, 4 | syl2anc 411 |
. 2
# # #
|
| 6 | sinadd 12247 |
. . . . 5
| |
| 7 | 6 | adantr 276 |
. . . 4
# # # |
| 8 | cosadd 12248 |
. . . . 5
| |
| 9 | 8 | adantr 276 |
. . . 4
# # # |
| 10 | 7, 9 | oveq12d 6019 |
. . 3
# # #
|
| 11 | simpll 527 |
. . . . . . . 8
# # # | |
| 12 | 11 | coscld 12222 |
. . . . . . 7
# # # |
| 13 | simplr 528 |
. . . . . . . 8
# # # | |
| 14 | 13 | coscld 12222 |
. . . . . . 7
# # # |
| 15 | 12, 14 | mulcld 8167 |
. . . . . 6
# # # |
| 16 | simpr1 1027 |
. . . . . . 7
# # # # | |
| 17 | 11, 16 | tanclapd 12223 |
. . . . . 6
# # # |
| 18 | simpr2 1028 |
. . . . . . 7
# # # # | |
| 19 | 13, 18 | tanclapd 12223 |
. . . . . 6
# # # |
| 20 | 15, 17, 19 | adddid 8171 |
. . . . 5
# # # |
| 21 | 12, 14, 17 | mul32d 8299 |
. . . . . . 7
# # # |
| 22 | tanvalap 12219 |
. . . . . . . . . . 11
# | |
| 23 | 11, 16, 22 | syl2anc 411 |
. . . . . . . . . 10
# # # |
| 24 | 23 | oveq2d 6017 |
. . . . . . . . 9
# # # |
| 25 | 11 | sincld 12221 |
. . . . . . . . . 10
# # # |
| 26 | 25, 12, 16 | divcanap2d 8939 |
. . . . . . . . 9
# # # |
| 27 | 24, 26 | eqtrd 2262 |
. . . . . . . 8
# # # |
| 28 | 27 | oveq1d 6016 |
. . . . . . 7
# # # |
| 29 | 21, 28 | eqtrd 2262 |
. . . . . 6
# # # |
| 30 | 12, 14, 19 | mulassd 8170 |
. . . . . . 7
# # # |
| 31 | tanvalap 12219 |
. . . . . . . . . . 11
# | |
| 32 | 13, 18, 31 | syl2anc 411 |
. . . . . . . . . 10
# # # |
| 33 | 32 | oveq2d 6017 |
. . . . . . . . 9
# # # |
| 34 | 13 | sincld 12221 |
. . . . . . . . . 10
# # # |
| 35 | 34, 14, 18 | divcanap2d 8939 |
. . . . . . . . 9
# # # |
| 36 | 33, 35 | eqtrd 2262 |
. . . . . . . 8
# # # |
| 37 | 36 | oveq2d 6017 |
. . . . . . 7
# # # |
| 38 | 30, 37 | eqtrd 2262 |
. . . . . 6
# # # |
| 39 | 29, 38 | oveq12d 6019 |
. . . . 5
# # # |
| 40 | 20, 39 | eqtrd 2262 |
. . . 4
# # # |
| 41 | 1cnd 8162 |
. . . . . 6
# # # | |
| 42 | 17, 19 | mulcld 8167 |
. . . . . 6
# # # |
| 43 | 15, 41, 42 | subdid 8560 |
. . . . 5
# # # |
| 44 | 15 | mulridd 8163 |
. . . . . 6
# # # |
| 45 | 12, 14, 17, 19 | mul4d 8301 |
. . . . . . 7
# # # |
| 46 | 27, 36 | oveq12d 6019 |
. . . . . . 7
# # # |
| 47 | 45, 46 | eqtrd 2262 |
. . . . . 6
# # # |
| 48 | 44, 47 | oveq12d 6019 |
. . . . 5
# # # |
| 49 | 43, 48 | eqtrd 2262 |
. . . 4
# # # |
| 50 | 40, 49 | oveq12d 6019 |
. . 3
# # # |
| 51 | 17, 19 | addcld 8166 |
. . . 4
# # # |
| 52 | ax-1cn 8092 |
. . . . 5
| |
| 53 | subcl 8345 |
. . . . 5
| |
| 54 | 52, 42, 53 | sylancr 414 |
. . . 4
# # # |
| 55 | tanaddaplem 12249 |
. . . . . . . 8
# # #  # | |
| 56 | 55 | 3adantr3 1182 |
. . . . . . 7
# # #
#
# |
| 57 | 3, 56 | mpbid 147 |
. . . . . 6
# # # # |
| 58 | apsym 8753 |
. . . . . . 7
# # | |
| 59 | 42, 41, 58 | syl2anc 411 |
. . . . . 6
# # # # # |
| 60 | 57, 59 | mpbid 147 |
. . . . 5
# # # # |
| 61 | 41, 42, 60 | subap0d 8791 |
. . . 4
# # # # |
| 62 | 12, 14, 16, 18 | mulap0d 8805 |
. . . 4
# # # # |
| 63 | 51, 54, 15, 61, 62 | divcanap5d 8964 |
. . 3
# # # |
| 64 | 10, 50, 63 | 3eqtr2d 2268 |
. 2
# # #
|
| 65 | 5, 64 | eqtrd 2262 |
1
# # # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 w3a 1002 wceq 1395 wcel 2200 class class class wbr 4083
cfv 5318
(class class class)co 6001 cc 7997 cc0 7999 c1 8000 caddc 8002 cmul 8004 cmin 8317 # cap 8728
cdiv 8819
csin 12155
ccos 12156
ctan 12157 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117 ax-arch 8118 ax-caucvg 8119 |
| This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-disj 4060 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-irdg 6516 df-frec 6537 df-1o 6562 df-oadd 6566 df-er 6680 df-en 6888 df-dom 6889 df-fin 6890 df-sup 7151 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-n0 9370 df-z 9447 df-uz 9723 df-q 9815 df-rp 9850 df-ico 10090 df-fz 10205 df-fzo 10339 df-seqfrec 10670 df-exp 10761 df-fac 10948 df-bc 10970 df-ihash 10998 df-cj 11353 df-re 11354 df-im 11355 df-rsqrt 11509 df-abs 11510 df-clim 11790 df-sumdc 11865 df-ef 12159 df-sin 12161 df-cos 12162 df-tan 12163 |
| This theorem is referenced by: (None) |
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