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Theorem negiso 9984
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 448 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR )  ->  x  e.  RR )
32renegcld 9464 . . . . . 6  |-  ( (  T.  /\  x  e.  RR )  ->  -u x  e.  RR )
4 simpr 448 . . . . . . 7  |-  ( (  T.  /\  y  e.  RR )  ->  y  e.  RR )
54renegcld 9464 . . . . . 6  |-  ( (  T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 recn 9080 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
7 recn 9080 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
8 negcon2 9354 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
96, 7, 8syl2an 464 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
109adantl 453 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  =  -u y 
<->  y  =  -u x
) )
111, 3, 5, 10f1ocnv2d 6295 . . . . 5  |-  (  T. 
->  ( F : RR -1-1-onto-> RR  /\  `' F  =  (
y  e.  RR  |->  -u y ) ) )
1211trud 1332 . . . 4  |-  ( F : RR -1-1-onto-> RR  /\  `' F  =  ( y  e.  RR  |->  -u y ) )
1312simpli 445 . . 3  |-  F : RR
-1-1-onto-> RR
14 ltneg 9528 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u y  <  -u z
) )
15 negex 9304 . . . . . . 7  |-  -u z  e.  _V
16 negex 9304 . . . . . . 7  |-  -u y  e.  _V
1715, 16brcnv 5055 . . . . . 6  |-  ( -u z `'  <  -u y  <->  -u y  <  -u z
)
1814, 17syl6bbr 255 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u z `'  <  -u y
) )
19 negeq 9298 . . . . . . 7  |-  ( x  =  z  ->  -u x  =  -u z )
2019, 1, 15fvmpt 5806 . . . . . 6  |-  ( z  e.  RR  ->  ( F `  z )  =  -u z )
21 negeq 9298 . . . . . . 7  |-  ( x  =  y  ->  -u x  =  -u y )
2221, 1, 16fvmpt 5806 . . . . . 6  |-  ( y  e.  RR  ->  ( F `  y )  =  -u y )
2320, 22breqan12d 4227 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( ( F `  z ) `'  <  ( F `  y )  <->  -u z `'  <  -u y
) )
2418, 23bitr4d 248 . . . 4  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  ( F `  z ) `'  <  ( F `  y ) ) )
2524rgen2a 2772 . . 3  |-  A. z  e.  RR  A. y  e.  RR  ( z  < 
y  <->  ( F `  z ) `'  <  ( F `  y ) )
26 df-isom 5463 . . 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 887 . 2  |-  F  Isom  <  ,  `'  <  ( RR ,  RR )
28 negeq 9298 . . . 4  |-  ( y  =  x  ->  -u y  =  -u x )
2928cbvmptv 4300 . . 3  |-  ( y  e.  RR  |->  -u y
)  =  ( x  e.  RR  |->  -u x
)
3012simpri 449 . . 3  |-  `' F  =  ( y  e.  RR  |->  -u y )
3129, 30, 13eqtr4i 2466 . 2  |-  `' F  =  F
3227, 31pm3.2i 442 1  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212    e. cmpt 4266   `'ccnv 4877   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455   CCcc 8988   RRcr 8989    < clt 9120   -ucneg 9292
This theorem is referenced by:  infmsup  9986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294
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