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Theorem recosf1o 23349
Description: The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.)
Assertion
Ref Expression
recosf1o  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> (
-u 1 [,] 1
)

Proof of Theorem recosf1o
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cosf 14157 . . . . . 6  |-  cos : CC
--> CC
2 ffn 5746 . . . . . 6  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
31, 2ax-mp 5 . . . . 5  |-  cos  Fn  CC
4 0re 9642 . . . . . . 7  |-  0  e.  RR
5 pire 23278 . . . . . . 7  |-  pi  e.  RR
6 iccssre 11716 . . . . . . 7  |-  ( ( 0  e.  RR  /\  pi  e.  RR )  -> 
( 0 [,] pi )  C_  RR )
74, 5, 6mp2an 676 . . . . . 6  |-  ( 0 [,] pi )  C_  RR
8 ax-resscn 9595 . . . . . 6  |-  RR  C_  CC
97, 8sstri 3479 . . . . 5  |-  ( 0 [,] pi )  C_  CC
10 fnssres 5707 . . . . 5  |-  ( ( cos  Fn  CC  /\  ( 0 [,] pi )  C_  CC )  -> 
( cos  |`  ( 0 [,] pi ) )  Fn  ( 0 [,] pi ) )
113, 9, 10mp2an 676 . . . 4  |-  ( cos  |`  ( 0 [,] pi ) )  Fn  (
0 [,] pi )
12 fvres 5895 . . . . . 6  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( cos `  x
) )
137sseli 3466 . . . . . . 7  |-  ( x  e.  ( 0 [,] pi )  ->  x  e.  RR )
14 cosbnd2 14215 . . . . . . 7  |-  ( x  e.  RR  ->  ( cos `  x )  e.  ( -u 1 [,] 1 ) )
1513, 14syl 17 . . . . . 6  |-  ( x  e.  ( 0 [,] pi )  ->  ( cos `  x )  e.  ( -u 1 [,] 1 ) )
1612, 15eqeltrd 2517 . . . . 5  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  e.  ( -u 1 [,] 1 ) )
1716rgen 2792 . . . 4  |-  A. x  e.  ( 0 [,] pi ) ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  e.  ( -u
1 [,] 1 )
18 ffnfv 6064 . . . 4  |-  ( ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u
1 [,] 1 )  <-> 
( ( cos  |`  (
0 [,] pi ) )  Fn  ( 0 [,] pi )  /\  A. x  e.  ( 0 [,] pi ) ( ( cos  |`  (
0 [,] pi ) ) `  x )  e.  ( -u 1 [,] 1 ) ) )
1911, 17, 18mpbir2an 928 . . 3  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20 fvres 5895 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 y )  =  ( cos `  y
) )
2112, 20eqeqan12d 2452 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  <-> 
( cos `  x
)  =  ( cos `  y ) ) )
22 cos11 23347 . . . . . 6  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( x  =  y  <->  ( cos `  x
)  =  ( cos `  y ) ) )
2322biimprd 226 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( cos `  x )  =  ( cos `  y )  ->  x  =  y ) )
2421, 23sylbid 218 . . . 4  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  ->  x  =  y ) )
2524rgen2a 2859 . . 3  |-  A. x  e.  ( 0 [,] pi ) A. y  e.  ( 0 [,] pi ) ( ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  ->  x  =  y )
26 dff13 6174 . . 3  |-  ( ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-> ( -u
1 [,] 1 )  <-> 
( ( cos  |`  (
0 [,] pi ) ) : ( 0 [,] pi ) --> (
-u 1 [,] 1
)  /\  A. x  e.  ( 0 [,] pi ) A. y  e.  ( 0 [,] pi ) ( ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  ->  x  =  y )
) )
2719, 25, 26mpbir2an 928 . 2  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi )
-1-1-> ( -u 1 [,] 1 )
284a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  0  e.  RR )
295a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  pi  e.  RR )
30 neg1rr 10714 . . . . . . . 8  |-  -u 1  e.  RR
31 1re 9641 . . . . . . . 8  |-  1  e.  RR
3230, 31elicc2i 11700 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
1 ) )
3332simp1bi 1020 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  e.  RR )
34 pipos 23280 . . . . . . 7  |-  0  <  pi
3534a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  0  <  pi )
369a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  (
0 [,] pi ) 
C_  CC )
37 coscn 23265 . . . . . . 7  |-  cos  e.  ( CC -cn-> CC )
3837a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  cos  e.  ( CC -cn-> CC ) )
397sseli 3466 . . . . . . . 8  |-  ( z  e.  ( 0 [,] pi )  ->  z  e.  RR )
4039recoscld 14176 . . . . . . 7  |-  ( z  e.  ( 0 [,] pi )  ->  ( cos `  z )  e.  RR )
4140adantl 467 . . . . . 6  |-  ( ( x  e.  ( -u
1 [,] 1 )  /\  z  e.  ( 0 [,] pi ) )  ->  ( cos `  z )  e.  RR )
42 cospi 23292 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
4332simp2bi 1021 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  -u 1  <_  x )
4442, 43syl5eqbr 4459 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  ( cos `  pi )  <_  x )
4532simp3bi 1022 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  1 )
46 cos0 14182 . . . . . . . 8  |-  ( cos `  0 )  =  1
4745, 46syl6breqr 4466 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  ( cos `  0
) )
4844, 47jca 534 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  (
( cos `  pi )  <_  x  /\  x  <_  ( cos `  0
) ) )
4928, 29, 33, 35, 36, 38, 41, 48ivthle2 22289 . . . . 5  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
50 eqcom 2438 . . . . . . 7  |-  ( x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  =  x )
5120eqeq1d 2431 . . . . . . 7  |-  ( y  e.  ( 0 [,] pi )  ->  (
( ( cos  |`  (
0 [,] pi ) ) `  y )  =  x  <->  ( cos `  y )  =  x ) )
5250, 51syl5bb 260 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
x  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  <->  ( cos `  y )  =  x ) )
5352rexbiia 2933 . . . . 5  |-  ( E. y  e.  ( 0 [,] pi ) x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
5449, 53sylibr 215 . . . 4  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y ) )
5554rgen 2792 . . 3  |-  A. x  e.  ( -u 1 [,] 1 ) E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y )
56 dffo3 6052 . . 3  |-  ( ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u
1 [,] 1 )  <-> 
( ( cos  |`  (
0 [,] pi ) ) : ( 0 [,] pi ) --> (
-u 1 [,] 1
)  /\  A. x  e.  ( -u 1 [,] 1 ) E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y ) ) )
5719, 55, 56mpbir2an 928 . 2  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi )
-onto-> ( -u 1 [,] 1 )
58 df-f1o 5608 . 2  |-  ( ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> ( -u 1 [,] 1 )  <->  ( ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-> ( -u 1 [,] 1 )  /\  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -onto-> ( -u 1 [,] 1 ) ) )
5927, 57, 58mpbir2an 928 1  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> (
-u 1 [,] 1
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783    C_ wss 3442   class class class wbr 4426    |` cres 4856    Fn wfn 5596   -->wf 5597   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    < clt 9674    <_ cle 9675   -ucneg 9860   [,]cicc 11638   cosccos 14095   picpi 14097   -cn->ccncf 21804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-pi 14104  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699
This theorem is referenced by:  resinf1o  23350
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