Theorem tanaddaplem 12298

Description: A useful intermediate step in tanaddap 12299 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 12267	. . . . 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 12267	. . . . 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 8199	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) e. CC )
6		sincl 12266	. . . . 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 12266	. . . . 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 8199	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( sin ` A ) x. ( sin ` B ) ) e. CC )
11		subap0 8822	. . 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 12297	. . . 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 4098	. 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 12268	. . . . . . 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 12268	. . . . . . 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 6035	. . . . 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 531	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` A ) # 0 )
22		simprr 533	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` B ) # 0 )
23	7, 2, 9, 4, 21, 22	divmuldivapd 9011	. . . . 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 2264	. . . 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 4098	. . 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 8194	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> 1 e. CC )
27	2, 4, 21, 22	mulap0d 8837	. . . 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 8992	. . 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 8195	. . . 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 4098	. . 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 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   - cmin 8349   # cap 8760   / cdiv 8851   sincsin 12204   cosccos 12205   tanctan 12206

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-ico 10128  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-fac 10987  df-bc 11009  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914  df-ef 12208  df-sin 12210  df-cos 12211  df-tan 12212

This theorem is referenced by:  tanaddap  12299