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Theorem negiso 9746
Description: Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
negiso.1  |-  F  =  ( x  e.  RR  |->  -u x )
Assertion
Ref Expression
negiso  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )

Proof of Theorem negiso
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negiso.1 . . . . . 6  |-  F  =  ( x  e.  RR  |->  -u x )
2 simpr 447 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR )  ->  x  e.  RR )
32renegcld 9226 . . . . . 6  |-  ( (  T.  /\  x  e.  RR )  ->  -u x  e.  RR )
4 simpr 447 . . . . . . 7  |-  ( (  T.  /\  y  e.  RR )  ->  y  e.  RR )
54renegcld 9226 . . . . . 6  |-  ( (  T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 recn 8843 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
7 recn 8843 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
8 negcon2 9116 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
96, 7, 8syl2an 463 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
109adantl 452 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  =  -u y 
<->  y  =  -u x
) )
111, 3, 5, 10f1ocnv2d 6084 . . . . 5  |-  (  T. 
->  ( F : RR -1-1-onto-> RR  /\  `' F  =  (
y  e.  RR  |->  -u y ) ) )
1211trud 1314 . . . 4  |-  ( F : RR -1-1-onto-> RR  /\  `' F  =  ( y  e.  RR  |->  -u y ) )
1312simpli 444 . . 3  |-  F : RR
-1-1-onto-> RR
14 ltneg 9290 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u y  <  -u z
) )
15 negex 9066 . . . . . . 7  |-  -u z  e.  _V
16 negex 9066 . . . . . . 7  |-  -u y  e.  _V
1715, 16brcnv 4880 . . . . . 6  |-  ( -u z `'  <  -u y  <->  -u y  <  -u z
)
1814, 17syl6bbr 254 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u z `'  <  -u y
) )
19 negeq 9060 . . . . . . 7  |-  ( x  =  z  ->  -u x  =  -u z )
2019, 1, 15fvmpt 5618 . . . . . 6  |-  ( z  e.  RR  ->  ( F `  z )  =  -u z )
21 negeq 9060 . . . . . . 7  |-  ( x  =  y  ->  -u x  =  -u y )
2221, 1, 16fvmpt 5618 . . . . . 6  |-  ( y  e.  RR  ->  ( F `  y )  =  -u y )
2320, 22breqan12d 4054 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( ( F `  z ) `'  <  ( F `  y )  <->  -u z `'  <  -u y
) )
2418, 23bitr4d 247 . . . 4  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  ( F `  z ) `'  <  ( F `  y ) ) )
2524rgen2a 2622 . . 3  |-  A. z  e.  RR  A. y  e.  RR  ( z  < 
y  <->  ( F `  z ) `'  <  ( F `  y ) )
26 df-isom 5280 . . 3  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  <-> 
( F : RR -1-1-onto-> RR  /\ 
A. z  e.  RR  A. y  e.  RR  (
z  <  y  <->  ( F `  z ) `'  <  ( F `  y ) ) ) )
2713, 25, 26mpbir2an 886 . 2  |-  F  Isom  <  ,  `'  <  ( RR ,  RR )
28 negeq 9060 . . . 4  |-  ( y  =  x  ->  -u y  =  -u x )
2928cbvmptv 4127 . . 3  |-  ( y  e.  RR  |->  -u y
)  =  ( x  e.  RR  |->  -u x
)
3012simpri 448 . . 3  |-  `' F  =  ( y  e.  RR  |->  -u y )
3129, 30, 13eqtr4i 2326 . 2  |-  `' F  =  F
3227, 31pm3.2i 441 1  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272   CCcc 8751   RRcr 8752    < clt 8883   -ucneg 9054
This theorem is referenced by:  infmsup  9748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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