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Theorem recosf1o 23484
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 14179 . . . . . 6  |-  cos : CC
--> CC
2 ffn 5728 . . . . . 6  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
31, 2ax-mp 5 . . . . 5  |-  cos  Fn  CC
4 0re 9643 . . . . . . 7  |-  0  e.  RR
5 pire 23413 . . . . . . 7  |-  pi  e.  RR
6 iccssre 11716 . . . . . . 7  |-  ( ( 0  e.  RR  /\  pi  e.  RR )  -> 
( 0 [,] pi )  C_  RR )
74, 5, 6mp2an 678 . . . . . 6  |-  ( 0 [,] pi )  C_  RR
8 ax-resscn 9596 . . . . . 6  |-  RR  C_  CC
97, 8sstri 3441 . . . . 5  |-  ( 0 [,] pi )  C_  CC
10 fnssres 5689 . . . . 5  |-  ( ( cos  Fn  CC  /\  ( 0 [,] pi )  C_  CC )  -> 
( cos  |`  ( 0 [,] pi ) )  Fn  ( 0 [,] pi ) )
113, 9, 10mp2an 678 . . . 4  |-  ( cos  |`  ( 0 [,] pi ) )  Fn  (
0 [,] pi )
12 fvres 5879 . . . . . 6  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( cos `  x
) )
137sseli 3428 . . . . . . 7  |-  ( x  e.  ( 0 [,] pi )  ->  x  e.  RR )
14 cosbnd2 14237 . . . . . . 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 2529 . . . . 5  |-  ( x  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 x )  e.  ( -u 1 [,] 1 ) )
1716rgen 2747 . . . 4  |-  A. x  e.  ( 0 [,] pi ) ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  e.  ( -u
1 [,] 1 )
18 ffnfv 6049 . . . 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 931 . . 3  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) --> ( -u 1 [,] 1 )
20 fvres 5879 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
( cos  |`  ( 0 [,] pi ) ) `
 y )  =  ( cos `  y
) )
2112, 20eqeqan12d 2467 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  <-> 
( cos `  x
)  =  ( cos `  y ) ) )
22 cos11 23482 . . . . . 6  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( x  =  y  <->  ( cos `  x
)  =  ( cos `  y ) ) )
2322biimprd 227 . . . . 5  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( cos `  x )  =  ( cos `  y )  ->  x  =  y ) )
2421, 23sylbid 219 . . . 4  |-  ( ( x  e.  ( 0 [,] pi )  /\  y  e.  ( 0 [,] pi ) )  ->  ( ( ( cos  |`  ( 0 [,] pi ) ) `
 x )  =  ( ( cos  |`  (
0 [,] pi ) ) `  y )  ->  x  =  y ) )
2524rgen2a 2815 . . 3  |-  A. x  e.  ( 0 [,] pi ) A. y  e.  ( 0 [,] pi ) ( ( ( cos  |`  ( 0 [,] pi ) ) `  x
)  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  ->  x  =  y )
26 dff13 6159 . . 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 931 . 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 9642 . . . . . . . 8  |-  1  e.  RR
3230, 31elicc2i 11700 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
1 ) )
3332simp1bi 1023 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  e.  RR )
34 pipos 23415 . . . . . . 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 23400 . . . . . . 7  |-  cos  e.  ( CC -cn-> CC )
3837a1i 11 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  cos  e.  ( CC -cn-> CC ) )
397sseli 3428 . . . . . . . 8  |-  ( z  e.  ( 0 [,] pi )  ->  z  e.  RR )
4039recoscld 14198 . . . . . . 7  |-  ( z  e.  ( 0 [,] pi )  ->  ( cos `  z )  e.  RR )
4140adantl 468 . . . . . 6  |-  ( ( x  e.  ( -u
1 [,] 1 )  /\  z  e.  ( 0 [,] pi ) )  ->  ( cos `  z )  e.  RR )
42 cospi 23427 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
4332simp2bi 1024 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  -u 1  <_  x )
4442, 43syl5eqbr 4436 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  ( cos `  pi )  <_  x )
4532simp3bi 1025 . . . . . . . 8  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  1 )
46 cos0 14204 . . . . . . . 8  |-  ( cos `  0 )  =  1
4745, 46syl6breqr 4443 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 1 )  ->  x  <_  ( cos `  0
) )
4844, 47jca 535 . . . . . 6  |-  ( x  e.  ( -u 1 [,] 1 )  ->  (
( cos `  pi )  <_  x  /\  x  <_  ( cos `  0
) ) )
4928, 29, 33, 35, 36, 38, 41, 48ivthle2 22408 . . . . 5  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
50 eqcom 2458 . . . . . . 7  |-  ( x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  =  x )
5120eqeq1d 2453 . . . . . . 7  |-  ( y  e.  ( 0 [,] pi )  ->  (
( ( cos  |`  (
0 [,] pi ) ) `  y )  =  x  <->  ( cos `  y )  =  x ) )
5250, 51syl5bb 261 . . . . . 6  |-  ( y  e.  ( 0 [,] pi )  ->  (
x  =  ( ( cos  |`  ( 0 [,] pi ) ) `
 y )  <->  ( cos `  y )  =  x ) )
5352rexbiia 2888 . . . . 5  |-  ( E. y  e.  ( 0 [,] pi ) x  =  ( ( cos  |`  ( 0 [,] pi ) ) `  y
)  <->  E. y  e.  ( 0 [,] pi ) ( cos `  y
)  =  x )
5449, 53sylibr 216 . . . 4  |-  ( x  e.  ( -u 1 [,] 1 )  ->  E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y ) )
5554rgen 2747 . . 3  |-  A. x  e.  ( -u 1 [,] 1 ) E. y  e.  ( 0 [,] pi ) x  =  (
( cos  |`  ( 0 [,] pi ) ) `
 y )
56 dffo3 6037 . . 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 931 . 2  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi )
-onto-> ( -u 1 [,] 1 )
58 df-f1o 5589 . 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 931 1  |-  ( cos  |`  ( 0 [,] pi ) ) : ( 0 [,] pi ) -1-1-onto-> (
-u 1 [,] 1
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738    C_ wss 3404   class class class wbr 4402    |` cres 4836    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    < clt 9675    <_ cle 9676   -ucneg 9861   [,]cicc 11638   cosccos 14117   picpi 14119   -cn->ccncf 21908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  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 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822
This theorem is referenced by:  resinf1o  23485
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