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Theorem negiso 9730
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 447 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR )  ->  x  e.  RR )
32renegcld 9210 . . . . . 6  |-  ( (  T.  /\  x  e.  RR )  ->  -u x  e.  RR )
4 simpr 447 . . . . . . 7  |-  ( (  T.  /\  y  e.  RR )  ->  y  e.  RR )
54renegcld 9210 . . . . . 6  |-  ( (  T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 recn 8827 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
7 recn 8827 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
8 negcon2 9100 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
96, 7, 8syl2an 463 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
109adantl 452 . . . . . 6  |-  ( (  T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  =  -u y 
<->  y  =  -u x
) )
111, 3, 5, 10f1ocnv2d 6068 . . . . 5  |-  (  T. 
->  ( F : RR -1-1-onto-> RR  /\  `' F  =  (
y  e.  RR  |->  -u y ) ) )
1211trud 1314 . . . 4  |-  ( F : RR -1-1-onto-> RR  /\  `' F  =  ( y  e.  RR  |->  -u y ) )
1312simpli 444 . . 3  |-  F : RR
-1-1-onto-> RR
14 ltneg 9274 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u y  <  -u z
) )
15 negex 9050 . . . . . . 7  |-  -u z  e.  _V
16 negex 9050 . . . . . . 7  |-  -u y  e.  _V
1715, 16brcnv 4864 . . . . . 6  |-  ( -u z `'  <  -u y  <->  -u y  <  -u z
)
1814, 17syl6bbr 254 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u z `'  <  -u y
) )
19 negeq 9044 . . . . . . 7  |-  ( x  =  z  ->  -u x  =  -u z )
2019, 1, 15fvmpt 5602 . . . . . 6  |-  ( z  e.  RR  ->  ( F `  z )  =  -u z )
21 negeq 9044 . . . . . . 7  |-  ( x  =  y  ->  -u x  =  -u y )
2221, 1, 16fvmpt 5602 . . . . . 6  |-  ( y  e.  RR  ->  ( F `  y )  =  -u y )
2320, 22breqan12d 4038 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( ( F `  z ) `'  <  ( F `  y )  <->  -u z `'  <  -u y
) )
2418, 23bitr4d 247 . . . 4  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  ( F `  z ) `'  <  ( F `  y ) ) )
2524rgen2a 2609 . . 3  |-  A. z  e.  RR  A. y  e.  RR  ( z  < 
y  <->  ( F `  z ) `'  <  ( F `  y ) )
26 df-isom 5264 . . 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 886 . 2  |-  F  Isom  <  ,  `'  <  ( RR ,  RR )
28 negeq 9044 . . . 4  |-  ( y  =  x  ->  -u y  =  -u x )
2928cbvmptv 4111 . . 3  |-  ( y  e.  RR  |->  -u y
)  =  ( x  e.  RR  |->  -u x
)
3012simpri 448 . . 3  |-  `' F  =  ( y  e.  RR  |->  -u y )
3129, 30, 13eqtr4i 2313 . 2  |-  `' F  =  F
3227, 31pm3.2i 441 1  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256   CCcc 8735   RRcr 8736    < clt 8867   -ucneg 9038
This theorem is referenced by:  infmsup  9732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040
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