MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  icoshftf1o Structured version   Unicode version

Theorem icoshftf1o 11639
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 11638 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
x  e.  ( A [,) B )  -> 
( x  +  C
)  e.  ( ( A  +  C ) [,) ( B  +  C ) ) ) )
21ralrimiv 2876 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A. x  e.  ( A [,) B
) ( x  +  C )  e.  ( ( A  +  C
) [,) ( B  +  C ) ) )
3 readdcl 9571 . . . . . . . . 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 9571 . . . . . . . . 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 9878 . . . . . . . . 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 11638 . . . . . . . 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 9639 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  RR* )
13 icossre 11601 . . . . . . . . . 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 3504 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  y  e.  RR )
1615recnd 9618 . . . . . . 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 9618 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  C  e.  CC )
1916, 18negsubd 9932 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  +  -u C )  =  ( y  -  C
) )
204recnd 9618 . . . . . . . . . 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 9618 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  CC )
2320, 22negsubd 9932 . . . . . . . . 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 9618 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  CC )
2625, 22pncand 9927 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  -  C )  =  A )
2723, 26eqtrd 2508 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  C
)  +  -u C
)  =  A )
286recnd 9618 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C )  e.  CC )
2928, 22negsubd 9932 . . . . . . . . 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 9618 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
3231, 22pncand 9927 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  -  C )  =  B )
3329, 32eqtrd 2508 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  +  C
)  +  -u C
)  =  B )
3427, 33oveq12d 6300 . . . . . . 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 2569 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  ( y  -  C )  e.  ( A [,) B ) )
37 reueq 3301 . . . . 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 9618 . . . . . . 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 9618 . . . . . . 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 9639 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  (
( A  +  C
) [,) ( B  +  C ) ) )  ->  B  e.  RR* )
46 icossre 11601 . . . . . . . . . 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 3504 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  y  e.  ( ( A  +  C ) [,) ( B  +  C )
) )  /\  x  e.  ( A [,) B
) )  ->  x  e.  RR )
4948recnd 9618 . . . . . . 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 9945 . . . . . 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 2476 . . . . . 6  |-  ( x  =  ( y  -  C )  <->  ( y  -  C )  =  x )
52 eqcom 2476 . . . . . 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 3045 . . . 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 2878 . 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 6041 . 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 1379    e. wcel 1767   A.wral 2814   E!wreu 2816    C_ wss 3476    |-> cmpt 4505   -1-1-onto->wf1o 5585  (class class class)co 6282   RRcr 9487    + caddc 9491   RR*cxr 9623    - cmin 9801   -ucneg 9802   [,)cico 11527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-ico 11531
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator