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Theorem iooref1o 15136
Description: A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)
Hypothesis
Ref Expression
iooref1o.f  |-  F  =  ( x  e.  RR  |->  ( 1  /  (
1  +  ( exp `  x ) ) ) )
Assertion
Ref Expression
iooref1o  |-  F : RR
-1-1-onto-> ( 0 (,) 1
)

Proof of Theorem iooref1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iooref1o.f . . 3  |-  F  =  ( x  e.  RR  |->  ( 1  /  (
1  +  ( exp `  x ) ) ) )
2 1rp 9671 . . . . . . . . 9  |-  1  e.  RR+
32a1i 9 . . . . . . . 8  |-  ( x  e.  RR  ->  1  e.  RR+ )
4 rpefcl 11707 . . . . . . . 8  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR+ )
53, 4rpaddcld 9726 . . . . . . 7  |-  ( x  e.  RR  ->  (
1  +  ( exp `  x ) )  e.  RR+ )
65rpreccld 9721 . . . . . 6  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  RR+ )
76rpred 9710 . . . . 5  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  RR )
86rpgt0d 9713 . . . . 5  |-  ( x  e.  RR  ->  0  <  ( 1  /  (
1  +  ( exp `  x ) ) ) )
9 1red 7986 . . . . . . 7  |-  ( x  e.  RR  ->  1  e.  RR )
109, 4ltaddrpd 9744 . . . . . 6  |-  ( x  e.  RR  ->  1  <  ( 1  +  ( exp `  x ) ) )
115recgt1d 9725 . . . . . 6  |-  ( x  e.  RR  ->  (
1  <  ( 1  +  ( exp `  x
) )  <->  ( 1  /  ( 1  +  ( exp `  x
) ) )  <  1 ) )
1210, 11mpbid 147 . . . . 5  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  <  1 )
13 0xr 8018 . . . . . 6  |-  0  e.  RR*
14 1re 7970 . . . . . . 7  |-  1  e.  RR
1514rexri 8029 . . . . . 6  |-  1  e.  RR*
16 elioo2 9935 . . . . . 6  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  (
( 1  /  (
1  +  ( exp `  x ) ) )  e.  ( 0 (,) 1 )  <->  ( (
1  /  ( 1  +  ( exp `  x
) ) )  e.  RR  /\  0  < 
( 1  /  (
1  +  ( exp `  x ) ) )  /\  ( 1  / 
( 1  +  ( exp `  x ) ) )  <  1
) ) )
1713, 15, 16mp2an 426 . . . . 5  |-  ( ( 1  /  ( 1  +  ( exp `  x
) ) )  e.  ( 0 (,) 1
)  <->  ( ( 1  /  ( 1  +  ( exp `  x
) ) )  e.  RR  /\  0  < 
( 1  /  (
1  +  ( exp `  x ) ) )  /\  ( 1  / 
( 1  +  ( exp `  x ) ) )  <  1
) )
187, 8, 12, 17syl3anbrc 1182 . . . 4  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  ( 0 (,) 1
) )
1918adantl 277 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  ( 0 (,) 1
) )
20 elioore 9926 . . . . . . . . . 10  |-  ( y  e.  ( 0 (,) 1 )  ->  y  e.  RR )
21 eliooord 9942 . . . . . . . . . . 11  |-  ( y  e.  ( 0 (,) 1 )  ->  (
0  <  y  /\  y  <  1 ) )
2221simpld 112 . . . . . . . . . 10  |-  ( y  e.  ( 0 (,) 1 )  ->  0  <  y )
2320, 22elrpd 9707 . . . . . . . . 9  |-  ( y  e.  ( 0 (,) 1 )  ->  y  e.  RR+ )
2423rpreccld 9721 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  (
1  /  y )  e.  RR+ )
2524rpred 9710 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  (
1  /  y )  e.  RR )
26 1red 7986 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  1  e.  RR )
2725, 26resubcld 8352 . . . . . 6  |-  ( y  e.  ( 0 (,) 1 )  ->  (
( 1  /  y
)  -  1 )  e.  RR )
2821simprd 114 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  y  <  1 )
2923reclt1d 9724 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  (
y  <  1  <->  1  <  ( 1  /  y ) ) )
3028, 29mpbid 147 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  1  <  ( 1  /  y
) )
3126, 25posdifd 8503 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  (
1  <  ( 1  /  y )  <->  0  <  ( ( 1  /  y
)  -  1 ) ) )
3230, 31mpbid 147 . . . . . 6  |-  ( y  e.  ( 0 (,) 1 )  ->  0  <  ( ( 1  / 
y )  -  1 ) )
3327, 32elrpd 9707 . . . . 5  |-  ( y  e.  ( 0 (,) 1 )  ->  (
( 1  /  y
)  -  1 )  e.  RR+ )
3433relogcld 14656 . . . 4  |-  ( y  e.  ( 0 (,) 1 )  ->  ( log `  ( ( 1  /  y )  - 
1 ) )  e.  RR )
3534adantl 277 . . 3  |-  ( ( T.  /\  y  e.  ( 0 (,) 1
) )  ->  ( log `  ( ( 1  /  y )  - 
1 ) )  e.  RR )
36 1cnd 7987 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  1  e.  CC )
374adantr 276 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( exp `  x
)  e.  RR+ )
3837rpcnd 9712 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( exp `  x
)  e.  CC )
3936, 38addcld 7991 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  +  ( exp `  x
) )  e.  CC )
4023adantl 277 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  y  e.  RR+ )
4140rpcnd 9712 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  y  e.  CC )
4240rpap0d 9716 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  y #  0 )
4336, 39, 41, 42divmulap2d 8795 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( 1  /  y )  =  ( 1  +  ( exp `  x ) )  <->  1  =  ( y  x.  ( 1  +  ( exp `  x
) ) ) ) )
4424adantl 277 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  / 
y )  e.  RR+ )
4544rpcnd 9712 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  / 
y )  e.  CC )
4636, 38, 45addrsub 8342 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( 1  +  ( exp `  x
) )  =  ( 1  /  y )  <-> 
( exp `  x
)  =  ( ( 1  /  y )  -  1 ) ) )
4733adantl 277 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( 1  /  y )  - 
1 )  e.  RR+ )
4847reeflogd 14657 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( exp `  ( log `  ( ( 1  /  y )  - 
1 ) ) )  =  ( ( 1  /  y )  - 
1 ) )
4948eqeq2d 2199 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( exp `  x )  =  ( exp `  ( log `  ( ( 1  / 
y )  -  1 ) ) )  <->  ( exp `  x )  =  ( ( 1  /  y
)  -  1 ) ) )
50 reef11 11721 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  ( log `  ( ( 1  /  y )  -  1 ) )  e.  RR )  -> 
( ( exp `  x
)  =  ( exp `  ( log `  (
( 1  /  y
)  -  1 ) ) )  <->  x  =  ( log `  ( ( 1  /  y )  -  1 ) ) ) )
5134, 50sylan2 286 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( exp `  x )  =  ( exp `  ( log `  ( ( 1  / 
y )  -  1 ) ) )  <->  x  =  ( log `  ( ( 1  /  y )  -  1 ) ) ) )
5246, 49, 513bitr2rd 217 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( x  =  ( log `  (
( 1  /  y
)  -  1 ) )  <->  ( 1  +  ( exp `  x
) )  =  ( 1  /  y ) ) )
53 eqcom 2189 . . . . . . 7  |-  ( ( 1  +  ( exp `  x ) )  =  ( 1  /  y
)  <->  ( 1  / 
y )  =  ( 1  +  ( exp `  x ) ) )
5452, 53bitrdi 196 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( x  =  ( log `  (
( 1  /  y
)  -  1 ) )  <->  ( 1  / 
y )  =  ( 1  +  ( exp `  x ) ) ) )
555adantr 276 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  +  ( exp `  x
) )  e.  RR+ )
5655rpap0d 9716 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  +  ( exp `  x
) ) #  0 )
5736, 41, 39, 56divmulap3d 8796 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( 1  /  ( 1  +  ( exp `  x
) ) )  =  y  <->  1  =  ( y  x.  ( 1  +  ( exp `  x
) ) ) ) )
5843, 54, 573bitr4d 220 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( x  =  ( log `  (
( 1  /  y
)  -  1 ) )  <->  ( 1  / 
( 1  +  ( exp `  x ) ) )  =  y ) )
59 eqcom 2189 . . . . 5  |-  ( ( 1  /  ( 1  +  ( exp `  x
) ) )  =  y  <->  y  =  ( 1  /  ( 1  +  ( exp `  x
) ) ) )
6058, 59bitrdi 196 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( x  =  ( log `  (
( 1  /  y
)  -  1 ) )  <->  y  =  ( 1  /  ( 1  +  ( exp `  x
) ) ) ) )
6160adantl 277 . . 3  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  ( 0 (,) 1
) ) )  -> 
( x  =  ( log `  ( ( 1  /  y )  -  1 ) )  <-> 
y  =  ( 1  /  ( 1  +  ( exp `  x
) ) ) ) )
621, 19, 35, 61f1o2d 6090 . 2  |-  ( T. 
->  F : RR -1-1-onto-> ( 0 (,) 1
) )
6362mptru 1372 1  |-  F : RR
-1-1-onto-> ( 0 (,) 1
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 979    = wceq 1363   T. wtru 1364    e. wcel 2158   class class class wbr 4015    |-> cmpt 4076   -1-1-onto->wf1o 5227   ` cfv 5228  (class class class)co 5888   RRcr 7824   0cc0 7825   1c1 7826    + caddc 7828    x. cmul 7830   RR*cxr 8005    < clt 8006    - cmin 8142    / cdiv 8643   RR+crp 9667   (,)cioo 9902   expce 11664   logclog 14630
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945  ax-pre-suploc 7946  ax-addf 7947  ax-mulf 7948
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-disj 3993  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-of 6097  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-oadd 6435  df-er 6549  df-map 6664  df-pm 6665  df-en 6755  df-dom 6756  df-fin 6757  df-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-xneg 9786  df-xadd 9787  df-ioo 9906  df-ico 9908  df-icc 9909  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-exp 10534  df-fac 10720  df-bc 10742  df-ihash 10770  df-shft 10838  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-sumdc 11376  df-ef 11670  df-e 11671  df-rest 12708  df-topgen 12727  df-psmet 13786  df-xmet 13787  df-met 13788  df-bl 13789  df-mopn 13790  df-top 13851  df-topon 13864  df-bases 13896  df-ntr 13949  df-cn 14041  df-cnp 14042  df-tx 14106  df-cncf 14411  df-limced 14478  df-dvap 14479  df-relog 14632
This theorem is referenced by:  iooreen  15137
  Copyright terms: Public domain W3C validator