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Theorem negiso 8937
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 8362 . . . . . 6  |-  ( ( T.  /\  x  e.  RR )  ->  -u x  e.  RR )
4 simpr 110 . . . . . . 7  |-  ( ( T.  /\  y  e.  RR )  ->  y  e.  RR )
54renegcld 8362 . . . . . 6  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 recn 7969 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
7 recn 7969 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
8 negcon2 8235 . . . . . . . 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 6094 . . . . 5  |-  ( T. 
->  ( F : RR -1-1-onto-> RR  /\  `' F  =  (
y  e.  RR  |->  -u y ) ) )
1211mptru 1373 . . . 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 8011 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  z  e.  CC )
1615negcld 8280 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  -> 
-u z  e.  CC )
177adantl 277 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
1817negcld 8280 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  -> 
-u y  e.  CC )
19 brcnvg 4823 . . . . . 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 8175 . . . . . . . 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 5614 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( F `  z
)  =  -u z
)
25 negeq 8175 . . . . . . . 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 5614 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( F `  y
)  =  -u y
)
2924, 28breq12d 4031 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( ( F `  z ) `'  <  ( F `  y )  <->  -u z `'  <  -u y
) )
30 ltneg 8444 . . . . 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 2544 . . 3  |-  A. z  e.  RR  A. y  e.  RR  ( z  < 
y  <->  ( F `  z ) `'  <  ( F `  y ) )
33 df-isom 5241 . . 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 944 . 2  |-  F  Isom  <  ,  `'  <  ( RR ,  RR )
35 negeq 8175 . . . 4  |-  ( y  =  x  ->  -u y  =  -u x )
3635cbvmptv 4114 . . 3  |-  ( y  e.  RR  |->  -u y
)  =  ( x  e.  RR  |->  -u x
)
3712simpri 113 . . 3  |-  `' F  =  ( y  e.  RR  |->  -u y )
3836, 37, 13eqtr4i 2220 . 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 1364   T. wtru 1365    e. wcel 2160   A.wral 2468   class class class wbr 4018    |-> cmpt 4079   `'ccnv 4640   -1-1-onto->wf1o 5231   ` cfv 5232    Isom wiso 5233   CCcc 7834   RRcr 7835    < clt 8017   -ucneg 8154
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-addass 7938  ax-distr 7940  ax-i2m1 7941  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947  ax-pre-ltadd 7952
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-isom 5241  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8019  df-mnf 8020  df-ltxr 8022  df-sub 8155  df-neg 8156
This theorem is referenced by:  infrenegsupex  9619
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