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Theorem icoshftf1o 11668
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  =  ( x  e.  ( A [,) B ) 
|->  ( x  +  C
) )
Assertion
Ref Expression
icoshftf1o  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  F : ( A [,) B ) -1-1-onto-> ( ( A  +  C ) [,) ( B  +  C )
) )
Distinct variable groups:    x, A    x, B    x, C
Allowed substitution hint:    F( x)

Proof of Theorem icoshftf1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 icoshft 11667 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
x  e.  ( A [,) B )  -> 
( x  +  C
)  e.  ( ( A  +  C ) [,) ( B  +  C ) ) ) )
21ralrimiv 2869 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A. x  e.  ( A [,) B
) ( x  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) ) )
3 readdcl 9592 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
433adant2 1015 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  RR )
5 readdcl 9592 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
653adant1 1014 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR )
7 renegcl 9901 . . . . . . . . 9  |-  ( C  e.  RR  ->  -u C  e.  RR )
873ad2ant3 1019 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  -u C  e.  RR )
9 icoshft 11667 . . . . . . . 8  |-  ( ( ( A  +  C
)  e.  RR  /\  ( B  +  C
)  e.  RR  /\  -u C  e.  RR )  ->  ( y  e.  ( ( A  +  C ) [,) ( B  +  C )
)  ->  ( y  +  -u C )  e.  ( ( ( A  +  C )  + 
-u C ) [,) ( ( B  +  C )  +  -u C ) ) ) )
104, 6, 8, 9syl3anc 1228 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
y  e.  ( ( A  +  C ) [,) ( B  +  C ) )  -> 
( y  +  -u C )  e.  ( ( ( A  +  C )  +  -u C ) [,) (
( B  +  C
)  +  -u C
) ) ) )
1110imp 429 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  +  -u C )  e.  ( ( ( A  +  C )  + 
-u C ) [,) ( ( B  +  C )  +  -u C ) ) )
126rexrd 9660 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR* )
13 icossre 11630 . . . . . . . . . 10  |-  ( ( ( A  +  C
)  e.  RR  /\  ( B  +  C
)  e.  RR* )  ->  ( ( A  +  C ) [,) ( B  +  C )
)  C_  RR )
144, 12, 13syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
) [,) ( B  +  C ) ) 
C_  RR )
1514sselda 3499 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  y  e.  RR )
1615recnd 9639 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  y  e.  CC )
17 simpl3 1001 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  C  e.  RR )
1817recnd 9639 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  C  e.  CC )
1916, 18negsubd 9956 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  +  -u C )  =  ( y  -  C
) )
204recnd 9639 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  CC )
21 simp3 998 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
2221recnd 9639 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  CC )
2320, 22negsubd 9956 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  +  -u C
)  =  ( ( A  +  C )  -  C ) )
24 simp1 996 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
2524recnd 9639 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  CC )
2625, 22pncand 9951 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  -  C )  =  A )
2723, 26eqtrd 2498 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  +  -u C
)  =  A )
286recnd 9639 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  CC )
2928, 22negsubd 9956 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  +  -u C
)  =  ( ( B  +  C )  -  C ) )
30 simp2 997 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
3130recnd 9639 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
3231, 22pncand 9951 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  -  C )  =  B )
3329, 32eqtrd 2498 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  +  -u C
)  =  B )
3427, 33oveq12d 6314 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( A  +  C )  +  -u C ) [,) (
( B  +  C
)  +  -u C
) )  =  ( A [,) B ) )
3534adantr 465 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( (
( A  +  C
)  +  -u C
) [,) ( ( B  +  C )  +  -u C ) )  =  ( A [,) B ) )
3611, 19, 353eltr3d 2559 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  -  C )  e.  ( A [,) B ) )
37 reueq 3297 . . . . 5  |-  ( ( y  -  C )  e.  ( A [,) B )  <->  E! x  e.  ( A [,) B
) x  =  ( y  -  C ) )
3836, 37sylib 196 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  E! x  e.  ( A [,) B
) x  =  ( y  -  C ) )
3915adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  y  e.  RR )
4039recnd 9639 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  y  e.  CC )
41 simpll3 1037 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  C  e.  RR )
4241recnd 9639 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  C  e.  CC )
43 simpl1 999 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  A  e.  RR )
44 simpl2 1000 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  B  e.  RR )
4544rexrd 9660 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  B  e.  RR* )
46 icossre 11630 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  C_  RR )
4743, 45, 46syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( A [,) B )  C_  RR )
4847sselda 3499 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  x  e.  RR )
4948recnd 9639 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  x  e.  CC )
5040, 42, 49subadd2d 9969 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  (
( y  -  C
)  =  x  <->  ( x  +  C )  =  y ) )
51 eqcom 2466 . . . . . 6  |-  ( x  =  ( y  -  C )  <->  ( y  -  C )  =  x )
52 eqcom 2466 . . . . . 6  |-  ( y  =  ( x  +  C )  <->  ( x  +  C )  =  y )
5350, 51, 523bitr4g 288 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  (
x  =  ( y  -  C )  <->  y  =  ( x  +  C
) ) )
5453reubidva 3041 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( E! x  e.  ( A [,) B ) x  =  ( y  -  C
)  <->  E! x  e.  ( A [,) B ) y  =  ( x  +  C ) ) )
5538, 54mpbid 210 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  E! x  e.  ( A [,) B
) y  =  ( x  +  C ) )
5655ralrimiva 2871 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A. y  e.  ( ( A  +  C ) [,) ( B  +  C )
) E! x  e.  ( A [,) B
) y  =  ( x  +  C ) )
57 icoshftf1o.1 . . 3  |-  F  =  ( x  e.  ( A [,) B ) 
|->  ( x  +  C
) )
5857f1ompt 6054 . 2  |-  ( F : ( A [,) B ) -1-1-onto-> ( ( A  +  C ) [,) ( B  +  C )
)  <->  ( A. x  e.  ( A [,) B
) ( x  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) )  /\  A. y  e.  ( ( A  +  C ) [,) ( B  +  C )
) E! x  e.  ( A [,) B
) y  =  ( x  +  C ) ) )
592, 56, 58sylanbrc 664 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  F : ( A [,) B ) -1-1-onto-> ( ( A  +  C ) [,) ( B  +  C )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E!wreu 2809    C_ wss 3471    |-> cmpt 4515   -1-1-onto->wf1o 5593  (class class class)co 6296   RRcr 9508    + caddc 9512   RR*cxr 9644    - cmin 9824   -ucneg 9825   [,)cico 11556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-ico 11560
This theorem is referenced by: (None)
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