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Theorem iccf1o 8622
Description: Describe a bijection from  0 ,  1 to an arbitrary nontrivial closed interval  , . (Contributed by Mario Carneiro, 8-Sep-2015.)
Hypothesis
Ref Expression
iccf1o.1  F  0 [,] 1 
|->  x.  +  1  -  x.
Assertion
Ref Expression
iccf1o  RR  RR  <  F : 0 [,] 1 -1-1-onto-> [,]  `' F  [,] 
|->  -  -
Distinct variable groups:   ,,   ,,
Allowed substitution hints:    F(,)

Proof of Theorem iccf1o
StepHypRef Expression
1 iccf1o.1 . 2  F  0 [,] 1 
|->  x.  +  1  -  x.
2 0re 6805 . . . . . . . . 9  0  RR
3 1re 6804 . . . . . . . . 9  1  RR
42, 3elicc2i 8558 . . . . . . . 8  0 [,] 1  RR  0  <_  <_ 
1
54simp1bi 918 . . . . . . 7  0 [,] 1  RR
65adantl 262 . . . . . 6  RR  RR  <  0 [,] 1  RR
76recnd 6831 . . . . 5  RR  RR  <  0 [,] 1  CC
8 simpl2 907 . . . . . 6  RR  RR  <  0 [,] 1  RR
98recnd 6831 . . . . 5  RR  RR  <  0 [,] 1  CC
107, 9mulcld 6825 . . . 4  RR  RR  <  0 [,] 1  x.  CC
11 ax-1cn 6756 . . . . . 6  1  CC
12 subcl 6987 . . . . . 6  1  CC  CC  1  -  CC
1311, 7, 12sylancr 393 . . . . 5  RR  RR  <  0 [,] 1  1  -  CC
14 simpl1 906 . . . . . 6  RR  RR  <  0 [,] 1  RR
1514recnd 6831 . . . . 5  RR  RR  <  0 [,] 1  CC
1613, 15mulcld 6825 . . . 4  RR  RR  <  0 [,] 1  1  -  x.  CC
1710, 16addcomd 6941 . . 3  RR  RR  <  0 [,] 1  x.  + 
1  -  x.  1  -  x.  +  x.
18 lincmb01cmp 8621 . . 3  RR  RR  <  0 [,] 1  1  -  x.  +  x.  [,]
1917, 18eqeltrd 2111 . 2  RR  RR  <  0 [,] 1  x.  + 
1  -  x.  [,]
20 simpr 103 . . . . 5  RR  RR  <  [,]  [,]
21 simpl1 906 . . . . . 6  RR  RR  <  [,]  RR
22 simpl2 907 . . . . . 6  RR  RR  <  [,]  RR
23 elicc2 8557 . . . . . . . . 9  RR  RR  [,]  RR  <_  <_
24233adant3 923 . . . . . . . 8  RR  RR  <  [,]  RR  <_  <_
2524biimpa 280 . . . . . . 7  RR  RR  <  [,]  RR  <_  <_
2625simp1d 915 . . . . . 6  RR  RR  <  [,]  RR
27 eqid 2037 . . . . . . 7  -  -
28 eqid 2037 . . . . . . 7  -  -
2927, 28iccshftl 8614 . . . . . 6  RR  RR  RR  RR  [,]  -  -  [,]  -
3021, 22, 26, 21, 29syl22anc 1135 . . . . 5  RR  RR  <  [,]  [,]  -  -  [,]  -
3120, 30mpbid 135 . . . 4  RR  RR  <  [,]  -  -  [,]  -
3226, 21resubcld 7155 . . . . . 6  RR  RR  <  [,]  -  RR
3332recnd 6831 . . . . 5  RR  RR  <  [,]  -  CC
34 difrp 8374 . . . . . . . 8  RR  RR  <  -  RR+
3534biimp3a 1234 . . . . . . 7  RR  RR  <  -  RR+
3635adantr 261 . . . . . 6  RR  RR  <  [,]  -  RR+
3736rpcnd 8379 . . . . 5  RR  RR  <  [,]  -  CC
38 rpap0 8354 . . . . . 6  -  RR+  - #  0
3936, 38syl 14 . . . . 5  RR  RR  <  [,]  - #  0
4033, 37, 39divcanap1d 7528 . . . 4  RR  RR  <  [,]  -  -  x.  -  -
4137mul02d 7165 . . . . . 6  RR  RR  <  [,]  0  x.  -  0
4221recnd 6831 . . . . . . 7  RR  RR  <  [,]  CC
4342subidd 7086 . . . . . 6  RR  RR  <  [,]  -  0
4441, 43eqtr4d 2072 . . . . 5  RR  RR  <  [,]  0  x.  -  -
4537mulid2d 6823 . . . . 5  RR  RR  <  [,]  1  x.  -  -
4644, 45oveq12d 5473 . . . 4  RR  RR  <  [,]  0  x.  -  [,] 1  x.  -  -  [,]  -
4731, 40, 463eltr4d 2118 . . 3  RR  RR  <  [,]  -  -  x.  -  0  x.  -  [,] 1  x.  -
48 0red 6806 . . . 4  RR  RR  <  [,]  0  RR
49 1red 6820 . . . 4  RR  RR  <  [,]  1  RR
5032, 36rerpdivcld 8404 . . . 4  RR  RR  <  [,]  -  -  RR
51 eqid 2037 . . . . 5  0  x.  -  0  x.  -
52 eqid 2037 . . . . 5  1  x.  -  1  x.  -
5351, 52iccdil 8616 . . . 4  0  RR  1  RR  -  -  RR  -  RR+  -  - 
0 [,] 1  -  -  x.  -  0  x.  -  [,]
1  x.  -
5448, 49, 50, 36, 53syl22anc 1135 . . 3  RR  RR  <  [,]  -  -  0 [,] 1  -  -  x.  -  0  x.  -  [,]
1  x.  -
5547, 54mpbird 156 . 2  RR  RR  <  [,]  -  -  0 [,] 1
56 eqcom 2039 . . . 4  -  -  -  -
5733adantrl 447 . . . . 5  RR  RR  < 
0 [,] 1  [,]  -  CC
587adantrr 448 . . . . 5  RR  RR  < 
0 [,] 1  [,]  CC
5937adantrl 447 . . . . 5  RR  RR  < 
0 [,] 1  [,]  -  CC
6039adantrl 447 . . . . 5  RR  RR  < 
0 [,] 1  [,]  - #  0
6157, 58, 59, 60divmulap3d 7561 . . . 4  RR  RR  < 
0 [,] 1  [,]  -  -  -  x.  -
6256, 61syl5bb 181 . . 3  RR  RR  < 
0 [,] 1  [,]  -  -  -  x.  -
6326adantrl 447 . . . . . 6  RR  RR  < 
0 [,] 1  [,]  RR
6463recnd 6831 . . . . 5  RR  RR  < 
0 [,] 1  [,]  CC
6542adantrl 447 . . . . 5  RR  RR  < 
0 [,] 1  [,]  CC
668, 14resubcld 7155 . . . . . . . 8  RR  RR  <  0 [,] 1  -  RR
676, 66remulcld 6833 . . . . . . 7  RR  RR  <  0 [,] 1  x.  -  RR
6867adantrr 448 . . . . . 6  RR  RR  < 
0 [,] 1  [,]  x.  -  RR
6968recnd 6831 . . . . 5  RR  RR  < 
0 [,] 1  [,]  x.  -  CC
7064, 65, 69subadd2d 7117 . . . 4  RR  RR  < 
0 [,] 1  [,]  -  x.  -  x.  -  +
71 eqcom 2039 . . . 4  x.  -  +  x.  -  +
7270, 71syl6bb 185 . . 3  RR  RR  < 
0 [,] 1  [,]  -  x.  -  x.  -  +
737, 15mulcld 6825 . . . . . . 7  RR  RR  <  0 [,] 1  x.  CC
7410, 73, 15subadd23d 7120 . . . . . 6  RR  RR  <  0 [,] 1  x.  -  x.  +  x.  +  -  x.
757, 9, 15subdid 7187 . . . . . . 7  RR  RR  <  0 [,] 1  x.  -  x.  -  x.
7675oveq1d 5470 . . . . . 6  RR  RR  <  0 [,] 1  x.  -  +  x.  -  x.  +
77 1cnd 6821 . . . . . . . . 9  RR  RR  <  0 [,] 1  1  CC
7877, 7, 15subdird 7188 . . . . . . . 8  RR  RR  <  0 [,] 1  1  -  x.  1  x.  -  x.
7915mulid2d 6823 . . . . . . . . 9  RR  RR  <  0 [,] 1  1  x.
8079oveq1d 5470 . . . . . . . 8  RR  RR  <  0 [,] 1  1  x.  -  x.  -  x.
8178, 80eqtrd 2069 . . . . . . 7  RR  RR  <  0 [,] 1  1  -  x.  -  x.
8281oveq2d 5471 . . . . . 6  RR  RR  <  0 [,] 1  x.  + 
1  -  x.  x.  +  -  x.
8374, 76, 823eqtr4d 2079 . . . . 5  RR  RR  <  0 [,] 1  x.  -  +  x.  +  1  -  x.
8483adantrr 448 . . . 4  RR  RR  < 
0 [,] 1  [,]  x.  -  +  x.  +  1  -  x.
8584eqeq2d 2048 . . 3  RR  RR  < 
0 [,] 1  [,]  x.  -  +  x.  +  1  -  x.
8662, 72, 853bitrd 203 . 2  RR  RR  < 
0 [,] 1  [,]  -  -  x.  + 
1  -  x.
871, 19, 55, 86f1ocnv2d 5646 1  RR  RR  <  F : 0 [,] 1 -1-1-onto-> [,]  `' F  [,] 
|->  -  -
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390   class class class wbr 3755    |-> cmpt 3809   `'ccnv 4287   -1-1-onto->wf1o 4844  (class class class)co 5455   CCcc 6689   RRcr 6690   0cc0 6691   1c1 6692    + caddc 6694    x. cmul 6696    < clt 6837    <_ cle 6838    - cmin 6959   # cap 7345   cdiv 7413   RR+crp 8338   [,]cicc 8510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6754  ax-resscn 6755  ax-1cn 6756  ax-1re 6757  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-mulrcl 6762  ax-addcom 6763  ax-mulcom 6764  ax-addass 6765  ax-mulass 6766  ax-distr 6767  ax-i2m1 6768  ax-1rid 6770  ax-0id 6771  ax-rnegex 6772  ax-precex 6773  ax-cnre 6774  ax-pre-ltirr 6775  ax-pre-ltwlin 6776  ax-pre-lttrn 6777  ax-pre-apti 6778  ax-pre-ltadd 6779  ax-pre-mulgt0 6780  ax-pre-mulext 6781
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-iltp 6452  df-enr 6634  df-nr 6635  df-ltr 6638  df-0r 6639  df-1r 6640  df-0 6698  df-1 6699  df-r 6701  df-lt 6704  df-pnf 6839  df-mnf 6840  df-xr 6841  df-ltxr 6842  df-le 6843  df-sub 6961  df-neg 6962  df-reap 7339  df-ap 7346  df-div 7414  df-rp 8339  df-icc 8514
This theorem is referenced by: (None)
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