Theorem tanaddaplem 11635

Description: A useful intermediate step in tanaddap 11636 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 11604	. . . . 5 |- ( A e. CC -> ( cos ` A ) e. CC )
2	1	ad2antrr 480	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` A ) e. CC )
3		coscl 11604	. . . . 5 |- ( B e. CC -> ( cos ` B ) e. CC )
4	3	ad2antlr 481	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` B ) e. CC )
5	2, 4	mulcld 7898	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) e. CC )
6		sincl 11603	. . . . 5 |- ( A e. CC -> ( sin ` A ) e. CC )
7	6	ad2antrr 480	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( sin ` A ) e. CC )
8		sincl 11603	. . . . 5 |- ( B e. CC -> ( sin ` B ) e. CC )
9	8	ad2antlr 481	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( sin ` B ) e. CC )
10	7, 9	mulcld 7898	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( sin ` A ) x. ( sin ` B ) ) e. CC )
11		subap0 8518	. . 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 409	. 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 11634	. . . 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 3975	. 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 11605	. . . . . . 7 |- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
17	16	ad2ant2r 501	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
18		tanvalap 11605	. . . . . . 7 |- ( ( B e. CC /\ ( cos ` B ) # 0 ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
19	18	ad2ant2l 500	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) )
20	17, 19	oveq12d 5842	. . . . 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 521	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` A ) # 0 )
22		simprr 522	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( cos ` B ) # 0 )
23	7, 2, 9, 4, 21, 22	divmuldivapd 8705	. . . . 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 2190	. . . 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 3975	. . 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 7894	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> 1 e. CC )
27	2, 4, 21, 22	mulap0d 8532	. . . 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 8686	. . 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 7895	. . . 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 3975	. . 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 1335   e. wcel 2128   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   0cc0 7732   1c1 7733   + caddc 7735   x. cmul 7737   - cmin 8046   # cap 8456   / cdiv 8545   sincsin 11541   cosccos 11542   tanctan 11543

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-disj 3943  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-frec 6338  df-1o 6363  df-oadd 6367  df-er 6480  df-en 6686  df-dom 6687  df-fin 6688  df-sup 6928  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-ico 9798  df-fz 9913  df-fzo 10042  df-seqfrec 10345  df-exp 10419  df-fac 10600  df-bc 10622  df-ihash 10650  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176  df-sumdc 11251  df-ef 11545  df-sin 11547  df-cos 11548  df-tan 11549

This theorem is referenced by:  tanaddap  11636