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Theorem reeff1o 21871
Description: The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
reeff1o  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+

Proof of Theorem reeff1o
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeff1 13400 . 2  |-  ( exp  |`  RR ) : RR -1-1-> RR+
2 f1f 5603 . . . 4  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
3 ffn 5556 . . . 4  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ( exp  |`  RR )  Fn  RR )
41, 2, 3mp2b 10 . . 3  |-  ( exp  |`  RR )  Fn  RR
5 frn 5562 . . . . 5  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ran  ( exp  |`  RR )  C_  RR+ )
61, 2, 5mp2b 10 . . . 4  |-  ran  ( exp  |`  RR )  C_  RR+
7 rpre 10993 . . . . . . . . 9  |-  ( z  e.  RR+  ->  z  e.  RR )
8 1re 9381 . . . . . . . . 9  |-  1  e.  RR
9 lttri4 9455 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
107, 8, 9sylancl 657 . . . . . . . 8  |-  ( z  e.  RR+  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
11 elrp 10989 . . . . . . . . . . . 12  |-  ( z  e.  RR+  <->  ( z  e.  RR  /\  0  < 
z ) )
12 reclt1 10223 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  0  <  z )  -> 
( z  <  1  <->  1  <  ( 1  / 
z ) ) )
1311, 12sylbi 195 . . . . . . . . . . 11  |-  ( z  e.  RR+  ->  ( z  <  1  <->  1  <  ( 1  /  z ) ) )
14 rpne0 11002 . . . . . . . . . . . . . . . 16  |-  ( z  e.  RR+  ->  z  =/=  0 )
157, 14rereccld 10154 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( 1  /  z )  e.  RR )
16 reeff1olem 21870 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  /  z
)  e.  RR  /\  1  <  ( 1  / 
z ) )  ->  E. y  e.  RR  ( exp `  y )  =  ( 1  / 
z ) )
1715, 16sylan 468 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. y  e.  RR  ( exp `  y
)  =  ( 1  /  z ) )
18 eqcom 2443 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  /  z )  =  ( exp `  y
)  <->  ( exp `  y
)  =  ( 1  /  z ) )
19 rpcnne0 11004 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  RR+  ->  ( z  e.  CC  /\  z  =/=  0 ) )
20 recn 9368 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  RR  ->  y  e.  CC )
21 efcl 13364 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2220, 21syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  ( exp `  y )  e.  CC )
23 efne0 13377 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  ( exp `  y )  =/=  0 )
2420, 23syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  ( exp `  y )  =/=  0 )
2522, 24jca 529 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
( exp `  y
)  e.  CC  /\  ( exp `  y )  =/=  0 ) )
26 rec11r 10026 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( z  e.  CC  /\  z  =/=  0 )  /\  ( ( exp `  y )  e.  CC  /\  ( exp `  y
)  =/=  0 ) )  ->  ( (
1  /  z )  =  ( exp `  y
)  <->  ( 1  / 
( exp `  y
) )  =  z ) )
2719, 25, 26syl2an 474 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( 1  /  ( exp `  y
) )  =  z ) )
28 efcan 13376 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  CC  ->  (
( exp `  y
)  x.  ( exp `  -u y ) )  =  1 )
2928eqcomd 2446 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  1  =  ( ( exp `  y )  x.  ( exp `  -u y ) ) )
30 negcl 9606 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  e.  CC  ->  -u y  e.  CC )
31 efcl 13364 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -u y  e.  CC  ->  ( exp `  -u y
)  e.  CC )
3230, 31syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  CC  ->  ( exp `  -u y )  e.  CC )
33 ax-1cn 9336 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  CC
34 divmul2 9994 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 1  e.  CC  /\  ( exp `  -u y
)  e.  CC  /\  ( ( exp `  y
)  e.  CC  /\  ( exp `  y )  =/=  0 ) )  ->  ( ( 1  /  ( exp `  y
) )  =  ( exp `  -u y
)  <->  1  =  ( ( exp `  y
)  x.  ( exp `  -u y ) ) ) )
3533, 34mp3an1 1296 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( exp `  -u y
)  e.  CC  /\  ( ( exp `  y
)  e.  CC  /\  ( exp `  y )  =/=  0 ) )  ->  ( ( 1  /  ( exp `  y
) )  =  ( exp `  -u y
)  <->  1  =  ( ( exp `  y
)  x.  ( exp `  -u y ) ) ) )
3632, 21, 23, 35syl12anc 1211 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  (
( 1  /  ( exp `  y ) )  =  ( exp `  -u y
)  <->  1  =  ( ( exp `  y
)  x.  ( exp `  -u y ) ) ) )
3729, 36mpbird 232 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  (
1  /  ( exp `  y ) )  =  ( exp `  -u y
) )
3820, 37syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  (
1  /  ( exp `  y ) )  =  ( exp `  -u y
) )
3938eqeq1d 2449 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
4039adantl 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
4127, 40bitrd 253 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( exp `  -u y )  =  z ) )
4218, 41syl5bbr 259 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( exp `  y
)  =  ( 1  /  z )  <->  ( exp `  -u y )  =  z ) )
4342biimpd 207 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( exp `  y
)  =  ( 1  /  z )  -> 
( exp `  -u y
)  =  z ) )
4443reximdva 2826 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( E. y  e.  RR  ( exp `  y )  =  ( 1  /  z
)  ->  E. y  e.  RR  ( exp `  -u y
)  =  z ) )
4544adantr 462 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  ( E. y  e.  RR  ( exp `  y )  =  ( 1  / 
z )  ->  E. y  e.  RR  ( exp `  -u y
)  =  z ) )
4617, 45mpd 15 . . . . . . . . . . . . 13  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. y  e.  RR  ( exp `  -u y
)  =  z )
47 renegcl 9668 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  -u y  e.  RR )
48 infm3lem 10284 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  E. y  e.  RR  x  =  -u y )
49 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( x  =  -u y  ->  ( exp `  x )  =  ( exp `  -u y
) )
5049eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( x  =  -u y  ->  (
( exp `  x
)  =  z  <->  ( exp `  -u y )  =  z ) )
5147, 48, 50rexxfr 4509 . . . . . . . . . . . . 13  |-  ( E. x  e.  RR  ( exp `  x )  =  z  <->  E. y  e.  RR  ( exp `  -u y
)  =  z )
5246, 51sylibr 212 . . . . . . . . . . . 12  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
5352ex 434 . . . . . . . . . . 11  |-  ( z  e.  RR+  ->  ( 1  <  ( 1  / 
z )  ->  E. x  e.  RR  ( exp `  x
)  =  z ) )
5413, 53sylbid 215 . . . . . . . . . 10  |-  ( z  e.  RR+  ->  ( z  <  1  ->  E. x  e.  RR  ( exp `  x
)  =  z ) )
5554imp 429 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  z  <  1 )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
56 ef0 13372 . . . . . . . . . . . 12  |-  ( exp `  0 )  =  1
5756eqeq2i 2451 . . . . . . . . . . 11  |-  ( z  =  ( exp `  0
)  <->  z  =  1 )
58 0re 9382 . . . . . . . . . . . . 13  |-  0  e.  RR
59 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
6059eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  (
( exp `  x
)  =  z  <->  ( exp `  0 )  =  z ) )
6160rspcev 3070 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  ( exp `  0 )  =  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6258, 61mpan 665 . . . . . . . . . . . 12  |-  ( ( exp `  0 )  =  z  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6362eqcoms 2444 . . . . . . . . . . 11  |-  ( z  =  ( exp `  0
)  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6457, 63sylbir 213 . . . . . . . . . 10  |-  ( z  =  1  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6564adantl 463 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  z  =  1 )  ->  E. x  e.  RR  ( exp `  x )  =  z )
66 reeff1olem 21870 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
677, 66sylan 468 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6855, 65, 673jaodan 1279 . . . . . . . 8  |-  ( ( z  e.  RR+  /\  (
z  <  1  \/  z  =  1  \/  1  <  z ) )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6910, 68mpdan 663 . . . . . . 7  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( exp `  x
)  =  z )
70 fvres 5701 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
( exp  |`  RR ) `
 x )  =  ( exp `  x
) )
7170eqeq1d 2449 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( ( exp  |`  RR ) `
 x )  =  z  <->  ( exp `  x
)  =  z ) )
7271rexbiia 2746 . . . . . . 7  |-  ( E. x  e.  RR  (
( exp  |`  RR ) `
 x )  =  z  <->  E. x  e.  RR  ( exp `  x )  =  z )
7369, 72sylibr 212 . . . . . 6  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( ( exp  |`  RR ) `  x
)  =  z )
74 fvelrnb 5736 . . . . . . 7  |-  ( ( exp  |`  RR )  Fn  RR  ->  ( z  e.  ran  ( exp  |`  RR )  <->  E. x  e.  RR  ( ( exp  |`  RR ) `
 x )  =  z ) )
754, 74ax-mp 5 . . . . . 6  |-  ( z  e.  ran  ( exp  |`  RR )  <->  E. x  e.  RR  ( ( exp  |`  RR ) `  x
)  =  z )
7673, 75sylibr 212 . . . . 5  |-  ( z  e.  RR+  ->  z  e. 
ran  ( exp  |`  RR ) )
7776ssriv 3357 . . . 4  |-  RR+  C_  ran  ( exp  |`  RR )
786, 77eqssi 3369 . . 3  |-  ran  ( exp  |`  RR )  = 
RR+
79 df-fo 5421 . . 3  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  <->  ( ( exp  |`  RR )  Fn  RR  /\ 
ran  ( exp  |`  RR )  =  RR+ ) )
804, 78, 79mpbir2an 906 . 2  |-  ( exp  |`  RR ) : RR -onto-> RR+
81 df-f1o 5422 . 2  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  <->  ( ( exp  |`  RR ) : RR -1-1-> RR+ 
/\  ( exp  |`  RR ) : RR -onto-> RR+ )
)
821, 80, 81mpbir2an 906 1  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 959    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    C_ wss 3325   class class class wbr 4289   ran crn 4837    |` cres 4838    Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    < clt 9414   -ucneg 9592    / cdiv 9989   RR+crp 10987   expce 13343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301
This theorem is referenced by:  reefiso  21872  efcvx  21873  reefgim  21874  eff1olem  21963  dfrelog  21976  relogf1o  21977  dvrelog  22041
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