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

Step	Hyp	Ref	Expression
1		coscl 11059	. . . . 5 |- ( A e. CC -> ( cos ` A ) e. CC )
2	1	ad2antrr 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 )
4	3	ad2antlr 474	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` B ) e. CC )
5	2, 4	mulcld 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 )
7	6	ad2antrr 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 )
9	8	ad2antlr 474	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( sin ` B ) e. CC )
10	7, 9	mulcld 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 ) ) ) )
12	5, 10, 11	syl2anc 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 ) ) ) )
14	13	adantr 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 ) ) ) )
15	14	breq1d 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 ) ) )
17	16	ad2ant2r 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 ) ) )
19	18	ad2ant2l 493	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
20	17, 19	oveq12d 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 )
23	7, 2, 9, 4, 21, 22	divmuldivapd 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 ) ) ) )
24	20, 23	eqtrd 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 ) ) ) )
25	24	breq1d 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 )
27	2, 4, 21, 22	mulap0d 8188	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) # 0 )
28	10, 5, 26, 27	apdivmuld 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 ) ) ) )
29	5	mulid1d 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 ) ) )
30	29	breq1d 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 ) ) ) )
31	25, 28, 30	3bitrd 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 ) ) ) )
32	12, 15, 31	3bitr4d 219	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( cos ` ( A + B ) ) # 0 <-> ( ( tan ` A ) x. ( tan ` B ) ) # 1 ) )

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