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Theorem negiso 10609
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 468 . . . . . . 7  |-  ( ( T.  /\  x  e.  RR )  ->  x  e.  RR )
32renegcld 10067 . . . . . 6  |-  ( ( T.  /\  x  e.  RR )  ->  -u x  e.  RR )
4 simpr 468 . . . . . . 7  |-  ( ( T.  /\  y  e.  RR )  ->  y  e.  RR )
54renegcld 10067 . . . . . 6  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 recn 9647 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
7 recn 9647 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
8 negcon2 9947 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
96, 7, 8syl2an 485 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
109adantl 473 . . . . . 6  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  =  -u y 
<->  y  =  -u x
) )
111, 3, 5, 10f1ocnv2d 6539 . . . . 5  |-  ( T. 
->  ( F : RR -1-1-onto-> RR  /\  `' F  =  (
y  e.  RR  |->  -u y ) ) )
1211trud 1461 . . . 4  |-  ( F : RR -1-1-onto-> RR  /\  `' F  =  ( y  e.  RR  |->  -u y ) )
1312simpli 465 . . 3  |-  F : RR
-1-1-onto-> RR
14 ltneg 10135 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u y  <  -u z
) )
15 negex 9893 . . . . . . 7  |-  -u z  e.  _V
16 negex 9893 . . . . . . 7  |-  -u y  e.  _V
1715, 16brcnv 5022 . . . . . 6  |-  ( -u z `'  <  -u y  <->  -u y  <  -u z
)
1814, 17syl6bbr 271 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u z `'  <  -u y
) )
19 negeq 9887 . . . . . . 7  |-  ( x  =  z  ->  -u x  =  -u z )
2019, 1, 15fvmpt 5963 . . . . . 6  |-  ( z  e.  RR  ->  ( F `  z )  =  -u z )
21 negeq 9887 . . . . . . 7  |-  ( x  =  y  ->  -u x  =  -u y )
2221, 1, 16fvmpt 5963 . . . . . 6  |-  ( y  e.  RR  ->  ( F `  y )  =  -u y )
2320, 22breqan12d 4411 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( ( F `  z ) `'  <  ( F `  y )  <->  -u z `'  <  -u y
) )
2418, 23bitr4d 264 . . . 4  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  ( F `  z ) `'  <  ( F `  y ) ) )
2524rgen2a 2820 . . 3  |-  A. z  e.  RR  A. y  e.  RR  ( z  < 
y  <->  ( F `  z ) `'  <  ( F `  y ) )
26 df-isom 5598 . . 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 934 . 2  |-  F  Isom  <  ,  `'  <  ( RR ,  RR )
28 negeq 9887 . . . 4  |-  ( y  =  x  ->  -u y  =  -u x )
2928cbvmptv 4488 . . 3  |-  ( y  e.  RR  |->  -u y
)  =  ( x  e.  RR  |->  -u x
)
3012simpri 469 . . 3  |-  `' F  =  ( y  e.  RR  |->  -u y )
3129, 30, 13eqtr4i 2503 . 2  |-  `' F  =  F
3227, 31pm3.2i 462 1  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   T. wtru 1453    e. wcel 1904   A.wral 2756   class class class wbr 4395    |-> cmpt 4454   `'ccnv 4838   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590   CCcc 9555   RRcr 9556    < clt 9693   -ucneg 9881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883
This theorem is referenced by:  infrenegsup  10613  infmsupOLD  10614
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