Theorem cos11 13414

Description: Cosine is one-to-one over the closed interval from 0 to pi. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Jim Kingdon, 6-May-2024.)

Assertion

Ref	Expression
cos11	|- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A = B <-> ( cos ` A ) = ( cos ` B ) ) )

Proof of Theorem cos11

Step	Hyp	Ref	Expression
1		fveq2 5486	. 2 |- ( A = B -> ( cos ` A ) = ( cos ` B ) )
2		simpll 519	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ A < B ) -> A e. ( 0 [,] pi ) )
3		simplr 520	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ A < B ) -> B e. ( 0 [,] pi ) )
4		simpr 109	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ A < B ) -> A < B )
5	2, 3, 4	cosordlem 13410	. . . . . . 7 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )
6	5	ex 114	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A < B -> ( cos ` B ) < ( cos ` A ) ) )
7		simplr 520	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ B < A ) -> B e. ( 0 [,] pi ) )
8		simpll 519	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ B < A ) -> A e. ( 0 [,] pi ) )
9		simpr 109	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ B < A ) -> B < A )
10	7, 8, 9	cosordlem 13410	. . . . . . 7 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ B < A ) -> ( cos ` A ) < ( cos ` B ) )
11	10	ex 114	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( B < A -> ( cos ` A ) < ( cos ` B ) ) )
12	6, 11	orim12d 776	. . . . 5 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( A < B \/ B < A ) -> ( ( cos ` B ) < ( cos ` A ) \/ ( cos ` A ) < ( cos ` B ) ) ) )
13		0re 7899	. . . . . . . . 9 |- 0 e. RR
14		pire 13347	. . . . . . . . 9 |- pi e. RR
15	13, 14	elicc2i 9875	. . . . . . . 8 |- ( A e. ( 0 [,] pi ) <-> ( A e. RR /\ 0 <_ A /\ A <_ pi ) )
16	15	simp1bi 1002	. . . . . . 7 |- ( A e. ( 0 [,] pi ) -> A e. RR )
17	16	adantr 274	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> A e. RR )
18	13, 14	elicc2i 9875	. . . . . . . 8 |- ( B e. ( 0 [,] pi ) <-> ( B e. RR /\ 0 <_ B /\ B <_ pi ) )
19	18	simp1bi 1002	. . . . . . 7 |- ( B e. ( 0 [,] pi ) -> B e. RR )
20	19	adantl 275	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> B e. RR )
21		reaplt 8486	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
22	17, 20, 21	syl2anc 409	. . . . 5 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A # B <-> ( A < B \/ B < A ) ) )
23	17	recoscld 11665	. . . . . . 7 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( cos ` A ) e. RR )
24	20	recoscld 11665	. . . . . . 7 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( cos ` B ) e. RR )
25		reaplt 8486	. . . . . . 7 |- ( ( ( cos ` A ) e. RR /\ ( cos ` B ) e. RR ) -> ( ( cos ` A ) # ( cos ` B ) <-> ( ( cos ` A ) < ( cos ` B ) \/ ( cos ` B ) < ( cos ` A ) ) ) )
26	23, 24, 25	syl2anc 409	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( cos ` A ) # ( cos ` B ) <-> ( ( cos ` A ) < ( cos ` B ) \/ ( cos ` B ) < ( cos ` A ) ) ) )
27		orcom 718	. . . . . 6 |- ( ( ( cos ` A ) < ( cos ` B ) \/ ( cos ` B ) < ( cos ` A ) ) <-> ( ( cos ` B ) < ( cos ` A ) \/ ( cos ` A ) < ( cos ` B ) ) )
28	26, 27	bitrdi 195	. . . . 5 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( cos ` A ) # ( cos ` B ) <-> ( ( cos ` B ) < ( cos ` A ) \/ ( cos ` A ) < ( cos ` B ) ) ) )
29	12, 22, 28	3imtr4d 202	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A # B -> ( cos ` A ) # ( cos ` B ) ) )
30	29	con3d 621	. . 3 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( -. ( cos ` A ) # ( cos ` B ) -> -. A # B ) )
31	23	recnd 7927	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( cos ` A ) e. CC )
32	24	recnd 7927	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( cos ` B ) e. CC )
33		apti 8520	. . . 4 |- ( ( ( cos ` A ) e. CC /\ ( cos ` B ) e. CC ) -> ( ( cos ` A ) = ( cos ` B ) <-> -. ( cos ` A ) # ( cos ` B ) ) )
34	31, 32, 33	syl2anc 409	. . 3 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( cos ` A ) = ( cos ` B ) <-> -. ( cos ` A ) # ( cos ` B ) ) )
35	17	recnd 7927	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> A e. CC )
36	20	recnd 7927	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> B e. CC )
37		apti 8520	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )
38	35, 36, 37	syl2anc 409	. . 3 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A = B <-> -. A # B ) )
39	30, 34, 38	3imtr4d 202	. 2 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( cos ` A ) = ( cos ` B ) -> A = B ) )
40	1, 39	impbid2 142	1 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A = B <-> ( cos ` A ) = ( cos ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   < clt 7933   <_ cle 7934   # cap 8479   [,]cicc 9827   cosccos 11586   picpi 11588

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-pre-suploc 7874  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-5 8919  df-6 8920  df-7 8921  df-8 8922  df-9 8923  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ioo 9828  df-ioc 9829  df-ico 9830  df-icc 9831  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-sin 11591  df-cos 11592  df-pi 11594  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by:  ioocosf1o  13415