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Theorem reeff1o 22028
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 13506 . 2  |-  ( exp  |`  RR ) : RR -1-1-> RR+
2 f1f 5704 . . . 4  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
3 ffn 5657 . . . 4  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ( exp  |`  RR )  Fn  RR )
41, 2, 3mp2b 10 . . 3  |-  ( exp  |`  RR )  Fn  RR
5 frn 5663 . . . . 5  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ran  ( exp  |`  RR )  C_  RR+ )
61, 2, 5mp2b 10 . . . 4  |-  ran  ( exp  |`  RR )  C_  RR+
7 rpre 11098 . . . . . . . . 9  |-  ( z  e.  RR+  ->  z  e.  RR )
8 1re 9486 . . . . . . . . 9  |-  1  e.  RR
9 lttri4 9560 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
107, 8, 9sylancl 662 . . . . . . . 8  |-  ( z  e.  RR+  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
11 elrp 11094 . . . . . . . . . . . 12  |-  ( z  e.  RR+  <->  ( z  e.  RR  /\  0  < 
z ) )
12 reclt1 10328 . . . . . . . . . . . 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 11107 . . . . . . . . . . . . . . . 16  |-  ( z  e.  RR+  ->  z  =/=  0 )
157, 14rereccld 10259 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( 1  /  z )  e.  RR )
16 reeff1olem 22027 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  /  z
)  e.  RR  /\  1  <  ( 1  / 
z ) )  ->  E. y  e.  RR  ( exp `  y )  =  ( 1  / 
z ) )
1715, 16sylan 471 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. y  e.  RR  ( exp `  y
)  =  ( 1  /  z ) )
18 eqcom 2460 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  /  z )  =  ( exp `  y
)  <->  ( exp `  y
)  =  ( 1  /  z ) )
19 rpcnne0 11109 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  RR+  ->  ( z  e.  CC  /\  z  =/=  0 ) )
20 recn 9473 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  RR  ->  y  e.  CC )
21 efcl 13470 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2220, 21syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  ( exp `  y )  e.  CC )
23 efne0 13483 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  ( exp `  y )  =/=  0 )
2420, 23syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  ( exp `  y )  =/=  0 )
2522, 24jca 532 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
( exp `  y
)  e.  CC  /\  ( exp `  y )  =/=  0 ) )
26 rec11r 10131 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( z  e.  CC  /\  z  =/=  0 )  /\  ( ( exp `  y )  e.  CC  /\  ( exp `  y
)  =/=  0 ) )  ->  ( (
1  /  z )  =  ( exp `  y
)  <->  ( 1  / 
( exp `  y
) )  =  z ) )
2719, 25, 26syl2an 477 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( 1  /  ( exp `  y
) )  =  z ) )
28 efcan 13482 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  CC  ->  (
( exp `  y
)  x.  ( exp `  -u y ) )  =  1 )
2928eqcomd 2459 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  1  =  ( ( exp `  y )  x.  ( exp `  -u y ) ) )
30 negcl 9711 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  e.  CC  ->  -u y  e.  CC )
31 efcl 13470 . . . . . . . . . . . . . . . . . . . . . . . . 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 9441 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  CC
34 divmul2 10099 . . . . . . . . . . . . . . . . . . . . . . . . 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 1302 . . . . . . . . . . . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . . . . . . . . . . . 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 2453 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
4039adantl 466 . . . . . . . . . . . . . . . . . . 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 2924 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( E. y  e.  RR  ( exp `  y )  =  ( 1  /  z
)  ->  E. y  e.  RR  ( exp `  -u y
)  =  z ) )
4544adantr 465 . . . . . . . . . . . . . 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 9773 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  -u y  e.  RR )
48 infm3lem 10389 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  E. y  e.  RR  x  =  -u y )
49 fveq2 5789 . . . . . . . . . . . . . . 15  |-  ( x  =  -u y  ->  ( exp `  x )  =  ( exp `  -u y
) )
5049eqeq1d 2453 . . . . . . . . . . . . . 14  |-  ( x  =  -u y  ->  (
( exp `  x
)  =  z  <->  ( exp `  -u y )  =  z ) )
5147, 48, 50rexxfr 4610 . . . . . . . . . . . . 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 13478 . . . . . . . . . . . 12  |-  ( exp `  0 )  =  1
5756eqeq2i 2469 . . . . . . . . . . 11  |-  ( z  =  ( exp `  0
)  <->  z  =  1 )
58 0re 9487 . . . . . . . . . . . . 13  |-  0  e.  RR
59 fveq2 5789 . . . . . . . . . . . . . . 15  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
6059eqeq1d 2453 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  (
( exp `  x
)  =  z  <->  ( exp `  0 )  =  z ) )
6160rspcev 3169 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  ( exp `  0 )  =  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6258, 61mpan 670 . . . . . . . . . . . 12  |-  ( ( exp `  0 )  =  z  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6362eqcoms 2463 . . . . . . . . . . 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 466 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  z  =  1 )  ->  E. x  e.  RR  ( exp `  x )  =  z )
66 reeff1olem 22027 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
677, 66sylan 471 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6855, 65, 673jaodan 1285 . . . . . . . 8  |-  ( ( z  e.  RR+  /\  (
z  <  1  \/  z  =  1  \/  1  <  z ) )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6910, 68mpdan 668 . . . . . . 7  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( exp `  x
)  =  z )
70 fvres 5803 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
( exp  |`  RR ) `
 x )  =  ( exp `  x
) )
7170eqeq1d 2453 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( ( exp  |`  RR ) `
 x )  =  z  <->  ( exp `  x
)  =  z ) )
7271rexbiia 2852 . . . . . . 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 5838 . . . . . . 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 3458 . . . 4  |-  RR+  C_  ran  ( exp  |`  RR )
786, 77eqssi 3470 . . 3  |-  ran  ( exp  |`  RR )  = 
RR+
79 df-fo 5522 . . 3  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  <->  ( ( exp  |`  RR )  Fn  RR  /\ 
ran  ( exp  |`  RR )  =  RR+ ) )
804, 78, 79mpbir2an 911 . 2  |-  ( exp  |`  RR ) : RR -onto-> RR+
81 df-f1o 5523 . 2  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  <->  ( ( exp  |`  RR ) : RR -1-1-> RR+ 
/\  ( exp  |`  RR ) : RR -onto-> RR+ )
)
821, 80, 81mpbir2an 911 1  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796    C_ wss 3426   class class class wbr 4390   ran crn 4939    |` cres 4940    Fn wfn 5511   -->wf 5512   -1-1->wf1 5513   -onto->wfo 5514   -1-1-onto->wf1o 5515   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384    x. cmul 9388    < clt 9519   -ucneg 9697    / cdiv 10094   RR+crp 11092   expce 13449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-ioo 11405  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-fac 12153  df-bc 12180  df-hash 12205  df-shft 12658  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-clim 13068  df-rlim 13069  df-sum 13266  df-ef 13455  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-fbas 17923  df-fg 17924  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cld 18739  df-ntr 18740  df-cls 18741  df-nei 18818  df-lp 18856  df-perf 18857  df-cn 18947  df-cnp 18948  df-haus 19035  df-tx 19251  df-hmeo 19444  df-fil 19535  df-fm 19627  df-flim 19628  df-flf 19629  df-xms 20011  df-ms 20012  df-tms 20013  df-cncf 20570  df-limc 21457  df-dv 21458
This theorem is referenced by:  reefiso  22029  efcvx  22030  reefgim  22031  eff1olem  22120  dfrelog  22133  relogf1o  22134  dvrelog  22198
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