Theorem cos11 14671

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 5530	. 2 |- ( A = B -> ( cos ` A ) = ( cos ` B ) )
2		simpll 527	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ A < B ) -> A e. ( 0 [,] pi ) )
3		simplr 528	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ A < B ) -> B e. ( 0 [,] pi ) )
4		simpr 110	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ A < B ) -> A < B )
5	2, 3, 4	cosordlem 14667	. . . . . . 7 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )
6	5	ex 115	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A < B -> ( cos ` B ) < ( cos ` A ) ) )
7		simplr 528	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ B < A ) -> B e. ( 0 [,] pi ) )
8		simpll 527	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ B < A ) -> A e. ( 0 [,] pi ) )
9		simpr 110	. . . . . . . 8 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ B < A ) -> B < A )
10	7, 8, 9	cosordlem 14667	. . . . . . 7 |- ( ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) /\ B < A ) -> ( cos ` A ) < ( cos ` B ) )
11	10	ex 115	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( B < A -> ( cos ` A ) < ( cos ` B ) ) )
12	6, 11	orim12d 787	. . . . 5 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( A < B \/ B < A ) -> ( ( cos ` B ) < ( cos ` A ) \/ ( cos ` A ) < ( cos ` B ) ) ) )
13		0re 7975	. . . . . . . . 9 |- 0 e. RR
14		pire 14604	. . . . . . . . 9 |- pi e. RR
15	13, 14	elicc2i 9957	. . . . . . . 8 |- ( A e. ( 0 [,] pi ) <-> ( A e. RR /\ 0 <_ A /\ A <_ pi ) )
16	15	simp1bi 1014	. . . . . . 7 |- ( A e. ( 0 [,] pi ) -> A e. RR )
17	16	adantr 276	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> A e. RR )
18	13, 14	elicc2i 9957	. . . . . . . 8 |- ( B e. ( 0 [,] pi ) <-> ( B e. RR /\ 0 <_ B /\ B <_ pi ) )
19	18	simp1bi 1014	. . . . . . 7 |- ( B e. ( 0 [,] pi ) -> B e. RR )
20	19	adantl 277	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> B e. RR )
21		reaplt 8563	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
22	17, 20, 21	syl2anc 411	. . . . 5 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A # B <-> ( A < B \/ B < A ) ) )
23	17	recoscld 11750	. . . . . . 7 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( cos ` A ) e. RR )
24	20	recoscld 11750	. . . . . . 7 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( cos ` B ) e. RR )
25		reaplt 8563	. . . . . . 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 411	. . . . . 6 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( cos ` A ) # ( cos ` B ) <-> ( ( cos ` A ) < ( cos ` B ) \/ ( cos ` B ) < ( cos ` A ) ) ) )
27		orcom 729	. . . . . 6 |- ( ( ( cos ` A ) < ( cos ` B ) \/ ( cos ` B ) < ( cos ` A ) ) <-> ( ( cos ` B ) < ( cos ` A ) \/ ( cos ` A ) < ( cos ` B ) ) )
28	26, 27	bitrdi 196	. . . . 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 203	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A # B -> ( cos ` A ) # ( cos ` B ) ) )
30	29	con3d 632	. . 3 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( -. ( cos ` A ) # ( cos ` B ) -> -. A # B ) )
31	23	recnd 8004	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( cos ` A ) e. CC )
32	24	recnd 8004	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( cos ` B ) e. CC )
33		apti 8597	. . . 4 |- ( ( ( cos ` A ) e. CC /\ ( cos ` B ) e. CC ) -> ( ( cos ` A ) = ( cos ` B ) <-> -. ( cos ` A ) # ( cos ` B ) ) )
34	31, 32, 33	syl2anc 411	. . 3 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( cos ` A ) = ( cos ` B ) <-> -. ( cos ` A ) # ( cos ` B ) ) )
35	17	recnd 8004	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> A e. CC )
36	20	recnd 8004	. . . 4 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> B e. CC )
37		apti 8597	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )
38	35, 36, 37	syl2anc 411	. . 3 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A = B <-> -. A # B ) )
39	30, 34, 38	3imtr4d 203	. 2 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( ( cos ` A ) = ( cos ` B ) -> A = B ) )
40	1, 39	impbid2 143	1 |- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A = B <-> ( cos ` A ) = ( cos ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2160   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   CCcc 7827   RRcr 7828   0cc0 7829   < clt 8010   <_ cle 8011   # cap 8556   [,]cicc 9909   cosccos 11671   picpi 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 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949  ax-pre-suploc 7950  ax-addf 7951  ax-mulf 7952

This theorem depends on definitions:  df-bi 117  df-stab 832  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-of 6101  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-map 6668  df-pm 6669  df-en 6759  df-dom 6760  df-fin 6761  df-sup 7001  df-inf 7002  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-5 8999  df-6 9000  df-7 9001  df-8 9002  df-9 9003  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-xneg 9790  df-xadd 9791  df-ioo 9910  df-ioc 9911  df-ico 9912  df-icc 9913  df-fz 10027  df-fzo 10161  df-seqfrec 10464  df-exp 10538  df-fac 10724  df-bc 10746  df-ihash 10774  df-shft 10842  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-clim 11305  df-sumdc 11380  df-ef 11674  df-sin 11676  df-cos 11677  df-pi 11679  df-rest 12712  df-topgen 12731  df-psmet 13817  df-xmet 13818  df-met 13819  df-bl 13820  df-mopn 13821  df-top 13895  df-topon 13908  df-bases 13940  df-ntr 13993  df-cn 14085  df-cnp 14086  df-tx 14150  df-cncf 14455  df-limced 14522  df-dvap 14523

This theorem is referenced by:  ioocosf1o  14672