MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tanaddlem Structured version   Unicode version

Theorem tanaddlem 13774
Description: A useful intermediate step in tanadd 13775 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.)
Assertion
Ref Expression
tanaddlem  |-  ( ( ( 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 tanaddlem
StepHypRef Expression
1 coscl 13735 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
21ad2antrr 725 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( cos `  A )  e.  CC )
3 coscl 13735 . . . . . 6  |-  ( B  e.  CC  ->  ( cos `  B )  e.  CC )
43ad2antlr 726 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( cos `  B )  e.  CC )
52, 4mulcld 9626 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( ( cos `  A )  x.  ( cos `  B
) )  e.  CC )
6 sincl 13734 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
76ad2antrr 725 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( sin `  A )  e.  CC )
8 sincl 13734 . . . . . 6  |-  ( B  e.  CC  ->  ( sin `  B )  e.  CC )
98ad2antlr 726 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( sin `  B )  e.  CC )
107, 9mulcld 9626 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( ( sin `  A )  x.  ( sin `  B
) )  e.  CC )
115, 10subeq0ad 9950 . . 3  |-  ( ( ( 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 ) ) ) )
12 cosadd 13773 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
1312adantr 465 . . . 4  |-  ( ( ( 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 ) ) ) )
1413eqeq1d 2469 . . 3  |-  ( ( ( 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 ) )
15 tanval 13736 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
1615ad2ant2r 746 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( tan `  A )  =  ( ( sin `  A
)  /  ( cos `  A ) ) )
17 tanval 13736 . . . . . . . 8  |-  ( ( B  e.  CC  /\  ( cos `  B )  =/=  0 )  -> 
( tan `  B
)  =  ( ( sin `  B )  /  ( cos `  B
) ) )
1817ad2ant2l 745 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( tan `  B )  =  ( ( sin `  B
)  /  ( cos `  B ) ) )
1916, 18oveq12d 6312 . . . . . 6  |-  ( ( ( 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 ) ) ) )
20 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( cos `  A )  =/=  0
)
21 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( cos `  B )  =/=  0
)
227, 2, 9, 4, 20, 21divmuldivd 10371 . . . . . 6  |-  ( ( ( 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 ) ) ) )
2319, 22eqtrd 2508 . . . . 5  |-  ( ( ( 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 ) ) ) )
2423eqeq1d 2469 . . . 4  |-  ( ( ( 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 ) )
25 1cnd 9622 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  1  e.  CC )
262, 4, 20, 21mulne0d 10211 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( ( cos `  A )  x.  ( cos `  B
) )  =/=  0
)
2710, 5, 25, 26divmuld 10352 . . . 4  |-  ( ( ( 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 ) ) ) )
285mulid1d 9623 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( (
( cos `  A
)  x.  ( cos `  B ) )  x.  1 )  =  ( ( cos `  A
)  x.  ( cos `  B ) ) )
2928eqeq1d 2469 . . . 4  |-  ( ( ( 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 ) ) ) )
3024, 27, 293bitrd 279 . . 3  |-  ( ( ( 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 ) ) ) )
3111, 14, 303bitr4d 285 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( cos `  A )  =/=  0  /\  ( cos `  B
)  =/=  0 ) )  ->  ( ( cos `  ( A  +  B ) )  =  0  <->  ( ( tan `  A )  x.  ( tan `  B ) )  =  1 ) )
3231necon3bid 2725 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5593  (class class class)co 6294   CCcc 9500   0cc0 9502   1c1 9503    + caddc 9505    x. cmul 9507    - cmin 9815    / cdiv 10216   sincsin 13673   cosccos 13674   tanctan 13675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580  ax-addf 9581  ax-mulf 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-ico 11545  df-fz 11683  df-fzo 11803  df-fl 11907  df-seq 12086  df-exp 12145  df-fac 12332  df-bc 12359  df-hash 12384  df-shft 12875  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-limsup 13269  df-clim 13286  df-rlim 13287  df-sum 13484  df-ef 13677  df-sin 13679  df-cos 13680  df-tan 13681
This theorem is referenced by:  tanadd  13775  tanregt0  22769
  Copyright terms: Public domain W3C validator