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Theorem negiso 8725
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 109 . . . . . . 7  |-  ( ( T.  /\  x  e.  RR )  ->  x  e.  RR )
32renegcld 8154 . . . . . 6  |-  ( ( T.  /\  x  e.  RR )  ->  -u x  e.  RR )
4 simpr 109 . . . . . . 7  |-  ( ( T.  /\  y  e.  RR )  ->  y  e.  RR )
54renegcld 8154 . . . . . 6  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 recn 7765 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
7 recn 7765 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
8 negcon2 8027 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
96, 7, 8syl2an 287 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
109adantl 275 . . . . . 6  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  =  -u y 
<->  y  =  -u x
) )
111, 3, 5, 10f1ocnv2d 5974 . . . . 5  |-  ( T. 
->  ( F : RR -1-1-onto-> RR  /\  `' F  =  (
y  e.  RR  |->  -u y ) ) )
1211mptru 1340 . . . 4  |-  ( F : RR -1-1-onto-> RR  /\  `' F  =  ( y  e.  RR  |->  -u y ) )
1312simpli 110 . . 3  |-  F : RR
-1-1-onto-> RR
14 simpl 108 . . . . . . . 8  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  z  e.  RR )
1514recnd 7806 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  z  e.  CC )
1615negcld 8072 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  -> 
-u z  e.  CC )
177adantl 275 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
1817negcld 8072 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  -> 
-u y  e.  CC )
19 brcnvg 4720 . . . . . 6  |-  ( (
-u z  e.  CC  /\  -u y  e.  CC )  ->  ( -u z `'  <  -u y  <->  -u y  <  -u z ) )
2016, 18, 19syl2anc 408 . . . . 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 7967 . . . . . . . 8  |-  ( x  =  z  ->  -u x  =  -u z )
2322adantl 275 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  y  e.  RR )  /\  x  =  z )  ->  -u x  = 
-u z )
2421, 23, 14, 16fvmptd 5502 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( F `  z
)  =  -u z
)
25 negeq 7967 . . . . . . . 8  |-  ( x  =  y  ->  -u x  =  -u y )
2625adantl 275 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  y  e.  RR )  /\  x  =  y )  ->  -u x  = 
-u y )
27 simpr 109 . . . . . . 7  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  y  e.  RR )
2821, 26, 27, 18fvmptd 5502 . . . . . 6  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( F `  y
)  =  -u y
)
2924, 28breq12d 3942 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( ( F `  z ) `'  <  ( F `  y )  <->  -u z `'  <  -u y
) )
30 ltneg 8236 . . . . 5  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  -u y  <  -u z
) )
3120, 29, 303bitr4rd 220 . . . 4  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  <->  ( F `  z ) `'  <  ( F `  y ) ) )
3231rgen2a 2486 . . 3  |-  A. z  e.  RR  A. y  e.  RR  ( z  < 
y  <->  ( F `  z ) `'  <  ( F `  y ) )
33 df-isom 5132 . . 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 926 . 2  |-  F  Isom  <  ,  `'  <  ( RR ,  RR )
35 negeq 7967 . . . 4  |-  ( y  =  x  ->  -u y  =  -u x )
3635cbvmptv 4024 . . 3  |-  ( y  e.  RR  |->  -u y
)  =  ( x  e.  RR  |->  -u x
)
3712simpri 112 . . 3  |-  `' F  =  ( y  e.  RR  |->  -u y )
3836, 37, 13eqtr4i 2170 . 2  |-  `' F  =  F
3934, 38pm3.2i 270 1  |-  ( F 
Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1331   T. wtru 1332    e. wcel 1480   A.wral 2416   class class class wbr 3929    |-> cmpt 3989   `'ccnv 4538   -1-1-onto->wf1o 5122   ` cfv 5123    Isom wiso 5124   CCcc 7630   RRcr 7631    < clt 7812   -ucneg 7946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-ltxr 7817  df-sub 7947  df-neg 7948
This theorem is referenced by:  infrenegsupex  9401
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