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Theorem negiso 8911
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 110 . . . . . . 7  |-  ( ( T.  /\  x  e.  RR )  ->  x  e.  RR )
32renegcld 8336 . . . . . 6  |-  ( ( T.  /\  x  e.  RR )  ->  -u x  e.  RR )
4 simpr 110 . . . . . . 7  |-  ( ( T.  /\  y  e.  RR )  ->  y  e.  RR )
54renegcld 8336 . . . . . 6  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 recn 7943 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
7 recn 7943 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
8 negcon2 8209 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
96, 7, 8syl2an 289 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
109adantl 277 . . . . . 6  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  =  -u y 
<->  y  =  -u x
) )
111, 3, 5, 10f1ocnv2d 6074 . . . . 5  |-  ( T. 
->  ( F : RR -1-1-onto-> RR  /\  `' F  =  (
y  e.  RR  |->  -u y ) ) )
1211mptru 1362 . . . 4  |-  ( F : RR -1-1-onto-> RR  /\  `' F  =  ( y  e.  RR  |->  -u y ) )
1312simpli 111 . . 3  |-  F : RR
-1-1-onto-> RR
14 simpl 109 . . . . . . . 8  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  z  e.  RR )
1514recnd 7985 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  z  e.  CC )
1615negcld 8254 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  -> 
-u z  e.  CC )
177adantl 277 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
1817negcld 8254 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  -> 
-u y  e.  CC )
19 brcnvg 4808 . . . . . 6  |-  ( (
-u z  e.  CC  /\  -u y  e.  CC )  ->  ( -u z `'  <  -u y  <->  -u y  <  -u z ) )
2016, 18, 19syl2anc 411 . . . . 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 8149 . . . . . . . 8  |-  ( x  =  z  ->  -u x  =  -u z )
2322adantl 277 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  y  e.  RR )  /\  x  =  z )  ->  -u x  = 
-u z )
2421, 23, 14, 16fvmptd 5597 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( F `  z
)  =  -u z
)
25 negeq 8149 . . . . . . . 8  |-  ( x  =  y  ->  -u x  =  -u y )
2625adantl 277 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  y  e.  RR )  /\  x  =  y )  ->  -u x  = 
-u y )
27 simpr 110 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  y  e.  RR )
2821, 26, 27, 18fvmptd 5597 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( F `  y
)  =  -u y
)
2924, 28breq12d 4016 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( ( F `  z ) `'  <  ( F `  y )  <->  -u z `'  <  -u y
) )
30 ltneg 8418 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u y  <  -u z
) )
3120, 29, 303bitr4rd 221 . . . 4  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  ( F `  z ) `'  <  ( F `  y ) ) )
3231rgen2a 2531 . . 3  |-  A. z  e.  RR  A. y  e.  RR  ( z  < 
y  <->  ( F `  z ) `'  <  ( F `  y ) )
33 df-isom 5225 . . 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 942 . 2  |-  F  Isom  <  ,  `'  <  ( RR ,  RR )
35 negeq 8149 . . . 4  |-  ( y  =  x  ->  -u y  =  -u x )
3635cbvmptv 4099 . . 3  |-  ( y  e.  RR  |->  -u y
)  =  ( x  e.  RR  |->  -u x
)
3712simpri 113 . . 3  |-  `' F  =  ( y  e.  RR  |->  -u y )
3836, 37, 13eqtr4i 2208 . 2  |-  `' F  =  F
3934, 38pm3.2i 272 1  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   T. wtru 1354    e. wcel 2148   A.wral 2455   class class class wbr 4003    |-> cmpt 4064   `'ccnv 4625   -1-1-onto->wf1o 5215   ` cfv 5216    Isom wiso 5217   CCcc 7808   RRcr 7809    < clt 7991   -ucneg 8128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-ltxr 7996  df-sub 8129  df-neg 8130
This theorem is referenced by:  infrenegsupex  9593
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