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Theorem negiso 8152
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 108 . . . . . . 7  |-  ( ( T.  /\  x  e.  RR )  ->  x  e.  RR )
32renegcld 7603 . . . . . 6  |-  ( ( T.  /\  x  e.  RR )  ->  -u x  e.  RR )
4 simpr 108 . . . . . . 7  |-  ( ( T.  /\  y  e.  RR )  ->  y  e.  RR )
54renegcld 7603 . . . . . 6  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 recn 7220 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
7 recn 7220 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
8 negcon2 7480 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
96, 7, 8syl2an 283 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
109adantl 271 . . . . . 6  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  =  -u y 
<->  y  =  -u x
) )
111, 3, 5, 10f1ocnv2d 5755 . . . . 5  |-  ( T. 
->  ( F : RR -1-1-onto-> RR  /\  `' F  =  (
y  e.  RR  |->  -u y ) ) )
1211trud 1294 . . . 4  |-  ( F : RR -1-1-onto-> RR  /\  `' F  =  ( y  e.  RR  |->  -u y ) )
1312simpli 109 . . 3  |-  F : RR
-1-1-onto-> RR
14 simpl 107 . . . . . . . 8  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  z  e.  RR )
1514recnd 7261 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  z  e.  CC )
1615negcld 7525 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  -> 
-u z  e.  CC )
177adantl 271 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
1817negcld 7525 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  -> 
-u y  e.  CC )
19 brcnvg 4564 . . . . . 6  |-  ( (
-u z  e.  CC  /\  -u y  e.  CC )  ->  ( -u z `'  <  -u y  <->  -u y  <  -u z ) )
2016, 18, 19syl2anc 403 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( -u z `'  <  -u y  <->  -u y  <  -u z ) )
211a1i 9 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  F  =  ( x  e.  RR  |->  -u x
) )
22 negeq 7420 . . . . . . . 8  |-  ( x  =  z  ->  -u x  =  -u z )
2322adantl 271 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  y  e.  RR )  /\  x  =  z )  ->  -u x  = 
-u z )
2421, 23, 14, 16fvmptd 5305 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( F `  z
)  =  -u z
)
25 negeq 7420 . . . . . . . 8  |-  ( x  =  y  ->  -u x  =  -u y )
2625adantl 271 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  y  e.  RR )  /\  x  =  y )  ->  -u x  = 
-u y )
27 simpr 108 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  y  e.  RR )
2821, 26, 27, 18fvmptd 5305 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( F `  y
)  =  -u y
)
2924, 28breq12d 3818 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( ( F `  z ) `'  <  ( F `  y )  <->  -u z `'  <  -u y
) )
30 ltneg 7685 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u y  <  -u z
) )
3120, 29, 303bitr4rd 219 . . . 4  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  ( F `  z ) `'  <  ( F `  y ) ) )
3231rgen2a 2422 . . 3  |-  A. z  e.  RR  A. y  e.  RR  ( z  < 
y  <->  ( F `  z ) `'  <  ( F `  y ) )
33 df-isom 4961 . . 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 ) ) ) )
3413, 32, 33mpbir2an 884 . 2  |-  F  Isom  <  ,  `'  <  ( RR ,  RR )
35 negeq 7420 . . . 4  |-  ( y  =  x  ->  -u y  =  -u x )
3635cbvmptv 3893 . . 3  |-  ( y  e.  RR  |->  -u y
)  =  ( x  e.  RR  |->  -u x
)
3712simpri 111 . . 3  |-  `' F  =  ( y  e.  RR  |->  -u y )
3836, 37, 13eqtr4i 2113 . 2  |-  `' F  =  F
3934, 38pm3.2i 266 1  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   T. wtru 1286    e. wcel 1434   A.wral 2353   class class class wbr 3805    |-> cmpt 3859   `'ccnv 4390   -1-1-onto->wf1o 4951   ` cfv 4952    Isom wiso 4953   CCcc 7093   RRcr 7094    < clt 7267   -ucneg 7399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-isom 4961  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-ltxr 7272  df-sub 7400  df-neg 7401
This theorem is referenced by:  infrenegsupex  8815
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