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Theorem tanaddaplem 11434
Description: A useful intermediate step in tanaddap 11435 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
Assertion
Ref Expression
tanaddaplem  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( cos `  ( A  +  B
) ) #  0  <->  (
( tan `  A
)  x.  ( tan `  B ) ) #  1 ) )

Proof of Theorem tanaddaplem
StepHypRef Expression
1 coscl 11403 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
21ad2antrr 479 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  A
)  e.  CC )
3 coscl 11403 . . . . 5  |-  ( B  e.  CC  ->  ( cos `  B )  e.  CC )
43ad2antlr 480 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  B
)  e.  CC )
52, 4mulcld 7779 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( cos `  A )  x.  ( cos `  B ) )  e.  CC )
6 sincl 11402 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
76ad2antrr 479 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( sin `  A
)  e.  CC )
8 sincl 11402 . . . . 5  |-  ( B  e.  CC  ->  ( sin `  B )  e.  CC )
98ad2antlr 480 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( sin `  B
)  e.  CC )
107, 9mulcld 7779 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( sin `  A )  x.  ( sin `  B ) )  e.  CC )
11 subap0 8398 . . 3  |-  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC  /\  ( ( sin `  A )  x.  ( sin `  B
) )  e.  CC )  ->  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) #  0  <->  ( ( cos `  A )  x.  ( cos `  B ) ) #  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
125, 10, 11syl2anc 408 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) #  0  <->  ( ( cos `  A )  x.  ( cos `  B ) ) #  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
13 cosadd 11433 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
1413adantr 274 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
1514breq1d 3934 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( cos `  ( A  +  B
) ) #  0  <->  (
( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) #  0 ) )
16 tanvalap 11404 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( tan `  A )  =  ( ( sin `  A
)  /  ( cos `  A ) ) )
1716ad2ant2r 500 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
18 tanvalap 11404 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( cos `  B ) #  0 )  ->  ( tan `  B )  =  ( ( sin `  B
)  /  ( cos `  B ) ) )
1918ad2ant2l 499 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( tan `  B
)  =  ( ( sin `  B )  /  ( cos `  B
) ) )
2017, 19oveq12d 5785 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( tan `  A )  x.  ( tan `  B ) )  =  ( ( ( sin `  A )  /  ( cos `  A
) )  x.  (
( sin `  B
)  /  ( cos `  B ) ) ) )
21 simprl 520 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  A
) #  0 )
22 simprr 521 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  B
) #  0 )
237, 2, 9, 4, 21, 22divmuldivapd 8585 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( ( sin `  A )  /  ( cos `  A
) )  x.  (
( sin `  B
)  /  ( cos `  B ) ) )  =  ( ( ( sin `  A )  x.  ( sin `  B
) )  /  (
( cos `  A
)  x.  ( cos `  B ) ) ) )
2420, 23eqtrd 2170 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( tan `  A )  x.  ( tan `  B ) )  =  ( ( ( sin `  A )  x.  ( sin `  B
) )  /  (
( cos `  A
)  x.  ( cos `  B ) ) ) )
2524breq1d 3934 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( ( tan `  A )  x.  ( tan `  B
) ) #  1  <->  (
( ( sin `  A
)  x.  ( sin `  B ) )  / 
( ( cos `  A
)  x.  ( cos `  B ) ) ) #  1 ) )
26 1cnd 7775 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  1  e.  CC )
272, 4, 21, 22mulap0d 8412 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( cos `  A )  x.  ( cos `  B ) ) #  0 )
2810, 5, 26, 27apdivmuld 8566 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( ( ( sin `  A
)  x.  ( sin `  B ) )  / 
( ( cos `  A
)  x.  ( cos `  B ) ) ) #  1  <->  ( ( ( cos `  A )  x.  ( cos `  B
) )  x.  1 ) #  ( ( sin `  A )  x.  ( sin `  B ) ) ) )
295mulid1d 7776 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( ( cos `  A )  x.  ( cos `  B
) )  x.  1 )  =  ( ( cos `  A )  x.  ( cos `  B
) ) )
3029breq1d 3934 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  x.  1 ) #  ( ( sin `  A )  x.  ( sin `  B
) )  <->  ( ( cos `  A )  x.  ( cos `  B
) ) #  ( ( sin `  A )  x.  ( sin `  B
) ) ) )
3125, 28, 303bitrd 213 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( ( tan `  A )  x.  ( tan `  B
) ) #  1  <->  (
( cos `  A
)  x.  ( cos `  B ) ) #  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
3212, 15, 313bitr4d 219 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( cos `  ( A  +  B
) ) #  0  <->  (
( tan `  A
)  x.  ( tan `  B ) ) #  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   0cc0 7613   1c1 7614    + caddc 7616    x. cmul 7618    - cmin 7926   # cap 8336    / cdiv 8425   sincsin 11339   cosccos 11340   tanctan 11341
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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-disj 3902  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-ico 9670  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-fac 10465  df-bc 10487  df-ihash 10515  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116  df-ef 11343  df-sin 11345  df-cos 11346  df-tan 11347
This theorem is referenced by:  tanaddap  11435
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