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Theorem icoshftf1o 10759
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 10758 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
x  e.  ( A [,) B )  -> 
( x  +  C
)  e.  ( ( A  +  C ) [,) ( B  +  C ) ) ) )
21ralrimiv 2625 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A. x  e.  ( A [,) B
) ( x  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) ) )
3 readdcl 8820 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
433adant2 974 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  RR )
5 readdcl 8820 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
653adant1 973 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR )
76rexrd 8881 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR* )
8 icossre 10730 . . . . . . . . . 10  |-  ( ( ( A  +  C
)  e.  RR  /\  ( B  +  C
)  e.  RR* )  ->  ( ( A  +  C ) [,) ( B  +  C )
)  C_  RR )
94, 7, 8syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
) [,) ( B  +  C ) ) 
C_  RR )
109sselda 3180 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  y  e.  RR )
1110recnd 8861 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  y  e.  CC )
12 simpl3 960 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  C  e.  RR )
1312recnd 8861 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  C  e.  CC )
1411, 13negsubd 9163 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  +  -u C )  =  ( y  -  C
) )
15 renegcl 9110 . . . . . . . . . 10  |-  ( C  e.  RR  ->  -u C  e.  RR )
16153ad2ant3 978 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  -u C  e.  RR )
17 icoshft 10758 . . . . . . . . 9  |-  ( ( ( 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 ) ) ) )
184, 6, 16, 17syl3anc 1182 . . . . . . . 8  |-  ( ( 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
) ) ) )
1918imp 418 . . . . . . 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 ) ) )
204recnd 8861 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  CC )
21 simp3 957 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
2221recnd 8861 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  CC )
2320, 22negsubd 9163 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  +  -u C
)  =  ( ( A  +  C )  -  C ) )
24 simp1 955 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
2524recnd 8861 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  CC )
2625, 22pncand 9158 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  -  C )  =  A )
2723, 26eqtrd 2315 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  +  -u C
)  =  A )
286recnd 8861 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  CC )
2928, 22negsubd 9163 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  +  -u C
)  =  ( ( B  +  C )  -  C ) )
30 simp2 956 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
3130recnd 8861 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
3231, 22pncand 9158 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  -  C )  =  B )
3329, 32eqtrd 2315 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  +  -u C
)  =  B )
3427, 33oveq12d 5876 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( A  +  C )  +  -u C ) [,) (
( B  +  C
)  +  -u C
) )  =  ( A [,) B ) )
3534adantr 451 . . . . . . 7  |-  ( ( ( 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 ) )
3619, 35eleqtrd 2359 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  +  -u C )  e.  ( A [,) B
) )
3714, 36eqeltrrd 2358 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  -  C )  e.  ( A [,) B ) )
38 reueq 2962 . . . . 5  |-  ( ( y  -  C )  e.  ( A [,) B )  <->  E! x  e.  ( A [,) B
) x  =  ( y  -  C ) )
3937, 38sylib 188 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  E! x  e.  ( A [,) B
) x  =  ( y  -  C ) )
4010adantr 451 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  y  e.  RR )
4140recnd 8861 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  y  e.  CC )
42 simpll3 996 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  C  e.  RR )
4342recnd 8861 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  C  e.  CC )
44 simpl1 958 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  A  e.  RR )
45 simpl2 959 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  B  e.  RR )
4645rexrd 8881 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  B  e.  RR* )
47 icossre 10730 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  C_  RR )
4844, 46, 47syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( A [,) B )  C_  RR )
4948sselda 3180 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  x  e.  RR )
5049recnd 8861 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  x  e.  CC )
5141, 43, 50subadd2d 9176 . . . . . 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 ) )
52 eqcom 2285 . . . . . 6  |-  ( x  =  ( y  -  C )  <->  ( y  -  C )  =  x )
53 eqcom 2285 . . . . . 6  |-  ( y  =  ( x  +  C )  <->  ( x  +  C )  =  y )
5451, 52, 533bitr4g 279 . . . . 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
) ) )
5554reubidva 2723 . . . 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 ) ) )
5639, 55mpbid 201 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  E! x  e.  ( A [,) B
) y  =  ( x  +  C ) )
5756ralrimiva 2626 . 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 ) )
58 icoshftf1o.1 . . 3  |-  F  =  ( x  e.  ( A [,) B ) 
|->  ( x  +  C
) )
5958f1ompt 5682 . 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 ) ) )
602, 57, 59sylanbrc 645 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  F : ( A [,) B ) -1-1-onto-> ( ( A  +  C ) [,) ( B  +  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E!wreu 2545    C_ wss 3152    e. cmpt 4077   -1-1-onto->wf1o 5254  (class class class)co 5858   RRcr 8736    + caddc 8740   RR*cxr 8866    - cmin 9037   -ucneg 9038   [,)cico 10658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-ico 10662
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