Theorem tanaddaplem 11765

Description: A useful intermediate step in tanaddap 11766 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 11734	. . . . 5 |- ( A e. CC -> ( cos ` A ) e. CC )
2	1	ad2antrr 488	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` A ) e. CC )
3		coscl 11734	. . . . 5 |- ( B e. CC -> ( cos ` B ) e. CC )
4	3	ad2antlr 489	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` B ) e. CC )
5	2, 4	mulcld 7997	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) e. CC )
6		sincl 11733	. . . . 5 |- ( A e. CC -> ( sin ` A ) e. CC )
7	6	ad2antrr 488	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( sin ` A ) e. CC )
8		sincl 11733	. . . . 5 |- ( B e. CC -> ( sin ` B ) e. CC )
9	8	ad2antlr 489	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( sin ` B ) e. CC )
10	7, 9	mulcld 7997	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( sin ` A ) x. ( sin ` B ) ) e. CC )
11		subap0 8619	. . 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 411	. 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 11764	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
14	13	adantr 276	. . 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 4028	. 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 11735	. . . . . . 7 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
17	16	ad2ant2r 509	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
18		tanvalap 11735	. . . . . . 7 |- ( ( B e. CC /\ ( cos ` B ) # 0 ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
19	18	ad2ant2l 508	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
20	17, 19	oveq12d 5909	. . . . 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 529	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` A ) # 0 )
22		simprr 531	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` B ) # 0 )
23	7, 2, 9, 4, 21, 22	divmuldivapd 8808	. . . . 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 2222	. . . 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 4028	. . 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 7992	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> 1 e. CC )
27	2, 4, 21, 22	mulap0d 8634	. . . 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 8789	. . 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	mulridd 7993	. . . 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 4028	. . 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 214	. 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 220	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 104   <-> wb 105   = wceq 1364   e. wcel 2160   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   CCcc 7828   0cc0 7830   1c1 7831   + caddc 7833   x. cmul 7835   - cmin 8147   # cap 8557   / cdiv 8648   sincsin 11671   cosccos 11672   tanctan 11673

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-disj 3996  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-frec 6410  df-1o 6435  df-oadd 6439  df-er 6553  df-en 6759  df-dom 6760  df-fin 6761  df-sup 7002  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-q 9639  df-rp 9673  df-ico 9913  df-fz 10028  df-fzo 10162  df-seqfrec 10465  df-exp 10539  df-fac 10725  df-bc 10747  df-ihash 10775  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027  df-clim 11306  df-sumdc 11381  df-ef 11675  df-sin 11677  df-cos 11678  df-tan 11679

This theorem is referenced by:  tanaddap  11766