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Theorem icoshftf1o 11518
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 11517 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
x  e.  ( A [,) B )  -> 
( x  +  C
)  e.  ( ( A  +  C ) [,) ( B  +  C ) ) ) )
21ralrimiv 2823 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A. x  e.  ( A [,) B
) ( x  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) ) )
3 readdcl 9469 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
433adant2 1007 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  RR )
5 readdcl 9469 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
653adant1 1006 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR )
7 renegcl 9776 . . . . . . . . 9  |-  ( C  e.  RR  ->  -u C  e.  RR )
873ad2ant3 1011 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  -u C  e.  RR )
9 icoshft 11517 . . . . . . . 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 1219 . . . . . . 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 9537 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR* )
13 icossre 11480 . . . . . . . . . 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 3457 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  y  e.  RR )
1615recnd 9516 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  y  e.  CC )
17 simpl3 993 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  C  e.  RR )
1817recnd 9516 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  C  e.  CC )
1916, 18negsubd 9829 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  +  -u C )  =  ( y  -  C
) )
204recnd 9516 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  +  C )  e.  CC )
21 simp3 990 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
2221recnd 9516 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  CC )
2320, 22negsubd 9829 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  +  -u C
)  =  ( ( A  +  C )  -  C ) )
24 simp1 988 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
2524recnd 9516 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  CC )
2625, 22pncand 9824 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  -  C )  =  A )
2723, 26eqtrd 2492 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  +  -u C
)  =  A )
286recnd 9516 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  CC )
2928, 22negsubd 9829 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  +  -u C
)  =  ( ( B  +  C )  -  C ) )
30 simp2 989 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
3130recnd 9516 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
3231, 22pncand 9824 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  -  C )  =  B )
3329, 32eqtrd 2492 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  +  -u C
)  =  B )
3427, 33oveq12d 6211 . . . . . . 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 2553 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  -  C )  e.  ( A [,) B ) )
37 reueq 3257 . . . . 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 9516 . . . . . . 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 1029 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  C  e.  RR )
4241recnd 9516 . . . . . . 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 991 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  A  e.  RR )
44 simpl2 992 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  B  e.  RR )
4544rexrd 9537 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  B  e.  RR* )
46 icossre 11480 . . . . . . . . . 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 3457 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  x  e.  RR )
4948recnd 9516 . . . . . . 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 9842 . . . . . 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 2460 . . . . . 6  |-  ( x  =  ( y  -  C )  <->  ( y  -  C )  =  x )
52 eqcom 2460 . . . . . 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 3003 . . . 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 2825 . 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 5967 . 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   E!wreu 2797    C_ wss 3429    |-> cmpt 4451   -1-1-onto->wf1o 5518  (class class class)co 6193   RRcr 9385    + caddc 9389   RR*cxr 9521    - cmin 9699   -ucneg 9700   [,)cico 11406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-ico 11410
This theorem is referenced by: (None)
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