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Theorem icoshftf1o 8629
Description: Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
Hypothesis
Ref Expression
icoshftf1o.1  F  [,) 
|->  +  C
Assertion
Ref Expression
icoshftf1o  RR  RR  C  RR  F : [,) -1-1-onto->  +  C [,)  +  C
Distinct variable groups:   ,   ,   , C
Allowed substitution hint:    F()

Proof of Theorem icoshftf1o
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 icoshft 8628 . . 3  RR  RR  C  RR  [,)  +  C  +  C [,)  +  C
21ralrimiv 2385 . 2  RR  RR  C  RR  [,)  +  C  +  C [,)  +  C
3 readdcl 6805 . . . . . . . . 9  RR  C  RR  +  C  RR
433adant2 922 . . . . . . . 8  RR  RR  C  RR  +  C  RR
5 readdcl 6805 . . . . . . . . 9  RR  C  RR  +  C  RR
653adant1 921 . . . . . . . 8  RR  RR  C  RR  +  C  RR
7 renegcl 7068 . . . . . . . . 9  C  RR  -u C  RR
873ad2ant3 926 . . . . . . . 8  RR  RR  C  RR  -u C  RR
9 icoshft 8628 . . . . . . . 8  +  C  RR  +  C  RR  -u C  RR  +  C [,)  +  C  +  -u C  +  C  +  -u C [,)  +  C  +  -u C
104, 6, 8, 9syl3anc 1134 . . . . . . 7  RR  RR  C  RR  +  C [,)  +  C  +  -u C  +  C  +  -u C [,)  +  C  +  -u C
1110imp 115 . . . . . 6  RR  RR  C  RR  +  C [,)  +  C  +  -u C  +  C  +  -u C [,)  +  C  +  -u C
126rexrd 6872 . . . . . . . . . 10  RR  RR  C  RR  +  C  RR*
13 icossre 8593 . . . . . . . . . 10  +  C  RR  +  C  RR*  +  C [,)  +  C  C_  RR
144, 12, 13syl2anc 391 . . . . . . . . 9  RR  RR  C  RR  +  C [,)  +  C  C_  RR
1514sselda 2939 . . . . . . . 8  RR  RR  C  RR  +  C [,)  +  C  RR
1615recnd 6851 . . . . . . 7  RR  RR  C  RR  +  C [,)  +  C  CC
17 simpl3 908 . . . . . . . 8  RR  RR  C  RR  +  C [,)  +  C  C  RR
1817recnd 6851 . . . . . . 7  RR  RR  C  RR  +  C [,)  +  C  C  CC
1916, 18negsubd 7124 . . . . . 6  RR  RR  C  RR  +  C [,)  +  C  +  -u C  -  C
204recnd 6851 . . . . . . . . . 10  RR  RR  C  RR  +  C  CC
21 simp3 905 . . . . . . . . . . 11  RR  RR  C  RR  C  RR
2221recnd 6851 . . . . . . . . . 10  RR  RR  C  RR  C  CC
2320, 22negsubd 7124 . . . . . . . . 9  RR  RR  C  RR  +  C  +  -u C  +  C  -  C
24 simp1 903 . . . . . . . . . . 11  RR  RR  C  RR  RR
2524recnd 6851 . . . . . . . . . 10  RR  RR  C  RR  CC
2625, 22pncand 7119 . . . . . . . . 9  RR  RR  C  RR  +  C  -  C
2723, 26eqtrd 2069 . . . . . . . 8  RR  RR  C  RR  +  C  +  -u C
286recnd 6851 . . . . . . . . . 10  RR  RR  C  RR  +  C  CC
2928, 22negsubd 7124 . . . . . . . . 9  RR  RR  C  RR  +  C  +  -u C  +  C  -  C
30 simp2 904 . . . . . . . . . . 11  RR  RR  C  RR  RR
3130recnd 6851 . . . . . . . . . 10  RR  RR  C  RR  CC
3231, 22pncand 7119 . . . . . . . . 9  RR  RR  C  RR  +  C  -  C
3329, 32eqtrd 2069 . . . . . . . 8  RR  RR  C  RR  +  C  +  -u C
3427, 33oveq12d 5473 . . . . . . 7  RR  RR  C  RR  +  C  +  -u C [,)  +  C  +  -u C  [,)
3534adantr 261 . . . . . 6  RR  RR  C  RR  +  C [,)  +  C  +  C  +  -u C [,)  +  C  +  -u C  [,)
3611, 19, 353eltr3d 2117 . . . . 5  RR  RR  C  RR  +  C [,)  +  C  -  C  [,)
37 reueq 2732 . . . . 5  -  C  [,)  [,)  -  C
3836, 37sylib 127 . . . 4  RR  RR  C  RR  +  C [,)  +  C  [,)  -  C
3915adantr 261 . . . . . . . 8  RR  RR  C  RR  +  C [,)  +  C  [,)  RR
4039recnd 6851 . . . . . . 7  RR  RR  C  RR  +  C [,)  +  C  [,)  CC
41 simpll3 944 . . . . . . . 8  RR  RR  C  RR  +  C [,)  +  C  [,)  C  RR
4241recnd 6851 . . . . . . 7  RR  RR  C  RR  +  C [,)  +  C  [,)  C  CC
43 simpl1 906 . . . . . . . . . 10  RR  RR  C  RR  +  C [,)  +  C  RR
44 simpl2 907 . . . . . . . . . . 11  RR  RR  C  RR  +  C [,)  +  C  RR
4544rexrd 6872 . . . . . . . . . 10  RR  RR  C  RR  +  C [,)  +  C  RR*
46 icossre 8593 . . . . . . . . . 10  RR  RR*  [,) 
C_  RR
4743, 45, 46syl2anc 391 . . . . . . . . 9  RR  RR  C  RR  +  C [,)  +  C  [,)  C_  RR
4847sselda 2939 . . . . . . . 8  RR  RR  C  RR  +  C [,)  +  C  [,)  RR
4948recnd 6851 . . . . . . 7  RR  RR  C  RR  +  C [,)  +  C  [,)  CC
5040, 42, 49subadd2d 7137 . . . . . 6  RR  RR  C  RR  +  C [,)  +  C  [,)  -  C  +  C
51 eqcom 2039 . . . . . 6  -  C  -  C
52 eqcom 2039 . . . . . 6  +  C  +  C
5350, 51, 523bitr4g 212 . . . . 5  RR  RR  C  RR  +  C [,)  +  C  [,)  -  C  +  C
5453reubidva 2486 . . . 4  RR  RR  C  RR  +  C [,)  +  C  [,)  -  C  [,)  +  C
5538, 54mpbid 135 . . 3  RR  RR  C  RR  +  C [,)  +  C  [,)  +  C
5655ralrimiva 2386 . 2  RR  RR  C  RR  +  C [,)  +  C  [,)  +  C
57 icoshftf1o.1 . . 3  F  [,) 
|->  +  C
5857f1ompt 5263 . 2  F : [,) -1-1-onto->  +  C [,)  +  C  [,)  +  C  +  C [,)  +  C  +  C [,)  +  C  [,)  +  C
592, 56, 58sylanbrc 394 1  RR  RR  C  RR  F : [,) -1-1-onto->  +  C [,)  +  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390  wral 2300  wreu 2302    C_ wss 2911    |-> cmpt 3809   -1-1-onto->wf1o 4844  (class class class)co 5455   RRcr 6710    + caddc 6714   RR*cxr 6856    - cmin 6979   -ucneg 6980   [,)cico 8529
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-bndl 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-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-ltadd 6799
This theorem depends on definitions:  df-bi 110  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-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-po 4024  df-iso 4025  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-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-ico 8533
This theorem is referenced by: (None)
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