Theorem tanaddaplem 11445

Description: A useful intermediate step in tanaddap 11446 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 11414	. . . . 5 |- ( A e. CC -> ( cos ` A ) e. CC )
2	1	ad2antrr 479	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` A ) e. CC )
3		coscl 11414	. . . . 5 |- ( B e. CC -> ( cos ` B ) e. CC )
4	3	ad2antlr 480	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` B ) e. CC )
5	2, 4	mulcld 7786	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) e. CC )
6		sincl 11413	. . . . 5 |- ( A e. CC -> ( sin ` A ) e. CC )
7	6	ad2antrr 479	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( sin ` A ) e. CC )
8		sincl 11413	. . . . 5 |- ( B e. CC -> ( sin ` B ) e. CC )
9	8	ad2antlr 480	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( sin ` B ) e. CC )
10	7, 9	mulcld 7786	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( sin ` A ) x. ( sin ` B ) ) e. CC )
11		subap0 8405	. . 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 408	. 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 11444	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
14	13	adantr 274	. . 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 3939	. 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 11415	. . . . . . 7 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
17	16	ad2ant2r 500	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
18		tanvalap 11415	. . . . . . 7 |- ( ( B e. CC /\ ( cos ` B ) # 0 ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
19	18	ad2ant2l 499	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
20	17, 19	oveq12d 5792	. . . . 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 520	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` A ) # 0 )
22		simprr 521	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` B ) # 0 )
23	7, 2, 9, 4, 21, 22	divmuldivapd 8592	. . . . 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 2172	. . . 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 3939	. . 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 7782	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> 1 e. CC )
27	2, 4, 21, 22	mulap0d 8419	. . . 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 8573	. . 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 7783	. . . 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 3939	. . 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 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   - cmin 7933   # cap 8343   / cdiv 8432   sincsin 11350   cosccos 11351   tanctan 11352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ico 9677  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-bc 10494  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123  df-ef 11354  df-sin 11356  df-cos 11357  df-tan 11358

This theorem is referenced by:  tanaddap  11446