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Theorem tanaddaplem 11090
Description: A useful intermediate step in tanaddap 11091 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 11059 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
21ad2antrr 473 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  A
)  e.  CC )
3 coscl 11059 . . . . 5  |-  ( B  e.  CC  ->  ( cos `  B )  e.  CC )
43ad2antlr 474 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  B
)  e.  CC )
52, 4mulcld 7569 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( cos `  A )  x.  ( cos `  B ) )  e.  CC )
6 sincl 11058 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
76ad2antrr 473 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( sin `  A
)  e.  CC )
8 sincl 11058 . . . . 5  |-  ( B  e.  CC  ->  ( sin `  B )  e.  CC )
98ad2antlr 474 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( sin `  B
)  e.  CC )
107, 9mulcld 7569 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( sin `  A )  x.  ( sin `  B ) )  e.  CC )
11 subap0 8179 . . 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 404 . 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 11089 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
1413adantr 271 . . 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 3861 . 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 11060 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A ) #  0 )  ->  ( tan `  A )  =  ( ( sin `  A
)  /  ( cos `  A ) ) )
1716ad2ant2r 494 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
18 tanvalap 11060 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( cos `  B ) #  0 )  ->  ( tan `  B )  =  ( ( sin `  B
)  /  ( cos `  B ) ) )
1918ad2ant2l 493 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( tan `  B
)  =  ( ( sin `  B )  /  ( cos `  B
) ) )
2017, 19oveq12d 5684 . . . . 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 499 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  A
) #  0 )
22 simprr 500 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( cos `  B
) #  0 )
237, 2, 9, 4, 21, 22divmuldivapd 8360 . . . . 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 2121 . . . 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 3861 . . 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 7565 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  1  e.  CC )
272, 4, 21, 22mulap0d 8188 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A ) #  0  /\  ( cos `  B
) #  0 ) )  ->  ( ( cos `  A )  x.  ( cos `  B ) ) #  0 )
2810, 5, 26, 27apdivmuld 8341 . . 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 7566 . . . 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 3861 . . 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 1290    e. wcel 1439   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   CCcc 7409   0cc0 7411   1c1 7412    + caddc 7414    x. cmul 7416    - cmin 7714   # cap 8119    / cdiv 8200   sincsin 10995   cosccos 10996   tanctan 10997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-disj 3829  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-sup 6733  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-ico 9373  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-fac 10195  df-bc 10217  df-ihash 10245  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804  df-ef 10999  df-sin 11001  df-cos 11002  df-tan 11003
This theorem is referenced by:  tanaddap  11091
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