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Theorem iooref1o 14323
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 9626 . . . . . . . . 9  |-  1  e.  RR+
32a1i 9 . . . . . . . 8  |-  ( x  e.  RR  ->  1  e.  RR+ )
4 rpefcl 11659 . . . . . . . 8  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR+ )
53, 4rpaddcld 9681 . . . . . . 7  |-  ( x  e.  RR  ->  (
1  +  ( exp `  x ) )  e.  RR+ )
65rpreccld 9676 . . . . . 6  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  RR+ )
76rpred 9665 . . . . 5  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  RR )
86rpgt0d 9668 . . . . 5  |-  ( x  e.  RR  ->  0  <  ( 1  /  (
1  +  ( exp `  x ) ) ) )
9 1red 7947 . . . . . . 7  |-  ( x  e.  RR  ->  1  e.  RR )
109, 4ltaddrpd 9699 . . . . . 6  |-  ( x  e.  RR  ->  1  <  ( 1  +  ( exp `  x ) ) )
115recgt1d 9680 . . . . . 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 7978 . . . . . 6  |-  0  e.  RR*
14 1re 7931 . . . . . . 7  |-  1  e.  RR
1514rexri 7989 . . . . . 6  |-  1  e.  RR*
16 elioo2 9890 . . . . . 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 1181 . . . 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 9881 . . . . . . . . . 10  |-  ( y  e.  ( 0 (,) 1 )  ->  y  e.  RR )
21 eliooord 9897 . . . . . . . . . . 11  |-  ( y  e.  ( 0 (,) 1 )  ->  (
0  <  y  /\  y  <  1 ) )
2221simpld 112 . . . . . . . . . 10  |-  ( y  e.  ( 0 (,) 1 )  ->  0  <  y )
2320, 22elrpd 9662 . . . . . . . . 9  |-  ( y  e.  ( 0 (,) 1 )  ->  y  e.  RR+ )
2423rpreccld 9676 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  (
1  /  y )  e.  RR+ )
2524rpred 9665 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  (
1  /  y )  e.  RR )
26 1red 7947 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  1  e.  RR )
2725, 26resubcld 8312 . . . . . 6  |-  ( y  e.  ( 0 (,) 1 )  ->  (
( 1  /  y
)  -  1 )  e.  RR )
2821simprd 114 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  y  <  1 )
2923reclt1d 9679 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  (
y  <  1  <->  1  <  ( 1  /  y ) ) )
3028, 29mpbid 147 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  1  <  ( 1  /  y
) )
3126, 25posdifd 8463 . . . . . . 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 9662 . . . . 5  |-  ( y  e.  ( 0 (,) 1 )  ->  (
( 1  /  y
)  -  1 )  e.  RR+ )
3433relogcld 13854 . . . 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 7948 . . . . . . 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 9667 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( exp `  x
)  e.  CC )
3936, 38addcld 7951 . . . . . . 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 9667 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  y  e.  CC )
4240rpap0d 9671 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  y #  0 )
4336, 39, 41, 42divmulap2d 8753 . . . . . 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 9667 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  / 
y )  e.  CC )
4636, 38, 45addrsub 8302 . . . . . . . 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 13855 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( exp `  ( log `  ( ( 1  /  y )  - 
1 ) ) )  =  ( ( 1  /  y )  - 
1 ) )
4948eqeq2d 2187 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( exp `  x )  =  ( exp `  ( log `  ( ( 1  / 
y )  -  1 ) ) )  <->  ( exp `  x )  =  ( ( 1  /  y
)  -  1 ) ) )
50 reef11 11673 . . . . . . . . 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 2177 . . . . . . 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 9671 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  +  ( exp `  x
) ) #  0 )
5736, 41, 39, 56divmulap3d 8754 . . . . . 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 2177 . . . . 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 6066 . 2  |-  ( T. 
->  F : RR -1-1-onto-> ( 0 (,) 1
) )
6362mptru 1362 1  |-  F : RR
-1-1-onto-> ( 0 (,) 1
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   T. wtru 1354    e. wcel 2146   class class class wbr 3998    |-> cmpt 4059   -1-1-onto->wf1o 5207   ` cfv 5208  (class class class)co 5865   RRcr 7785   0cc0 7786   1c1 7787    + caddc 7789    x. cmul 7791   RR*cxr 7965    < clt 7966    - cmin 8102    / cdiv 8601   RR+crp 9622   (,)cioo 9857   expce 11616   logclog 13828
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906  ax-pre-suploc 7907  ax-addf 7908  ax-mulf 7909
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-disj 3976  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-of 6073  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-oadd 6411  df-er 6525  df-map 6640  df-pm 6641  df-en 6731  df-dom 6732  df-fin 6733  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-xneg 9741  df-xadd 9742  df-ioo 9861  df-ico 9863  df-icc 9864  df-fz 9978  df-fzo 10111  df-seqfrec 10414  df-exp 10488  df-fac 10672  df-bc 10694  df-ihash 10722  df-shft 10790  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-clim 11253  df-sumdc 11328  df-ef 11622  df-e 11623  df-rest 12610  df-topgen 12629  df-psmet 13038  df-xmet 13039  df-met 13040  df-bl 13041  df-mopn 13042  df-top 13047  df-topon 13060  df-bases 13092  df-ntr 13147  df-cn 13239  df-cnp 13240  df-tx 13304  df-cncf 13609  df-limced 13676  df-dvap 13677  df-relog 13830
This theorem is referenced by:  iooreen  14324
  Copyright terms: Public domain W3C validator