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Theorem negf1o 8056
Description: Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.)
Hypothesis
Ref Expression
negf1o.1  |-  F  =  ( x  e.  A  |-> 
-u x )
Assertion
Ref Expression
negf1o  |-  ( A 
C_  RR  ->  F : A
-1-1-onto-> { n  e.  RR  |  -u n  e.  A } )
Distinct variable group:    A, n, x
Allowed substitution hints:    F( x, n)

Proof of Theorem negf1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 negf1o.1 . . 3  |-  F  =  ( x  e.  A  |-> 
-u x )
2 ssel 3055 . . . . . 6  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
3 renegcl 7939 . . . . . 6  |-  ( x  e.  RR  ->  -u x  e.  RR )
42, 3syl6 33 . . . . 5  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  -u x  e.  RR ) )
54imp 123 . . . 4  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  -u x  e.  RR )
62imp 123 . . . . 5  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
7 recn 7670 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
8 negneg 7928 . . . . . . . . . 10  |-  ( x  e.  CC  ->  -u -u x  =  x )
98eqcomd 2118 . . . . . . . . 9  |-  ( x  e.  CC  ->  x  =  -u -u x )
107, 9syl 14 . . . . . . . 8  |-  ( x  e.  RR  ->  x  =  -u -u x )
1110eleq1d 2181 . . . . . . 7  |-  ( x  e.  RR  ->  (
x  e.  A  <->  -u -u x  e.  A ) )
1211biimpcd 158 . . . . . 6  |-  ( x  e.  A  ->  (
x  e.  RR  ->  -u -u x  e.  A ) )
1312adantl 273 . . . . 5  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  (
x  e.  RR  ->  -u -u x  e.  A ) )
146, 13mpd 13 . . . 4  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  -u -u x  e.  A )
15 negeq 7871 . . . . . 6  |-  ( n  =  -u x  ->  -u n  =  -u -u x )
1615eleq1d 2181 . . . . 5  |-  ( n  =  -u x  ->  ( -u n  e.  A  <->  -u -u x  e.  A ) )
1716elrab 2807 . . . 4  |-  ( -u x  e.  { n  e.  RR  |  -u n  e.  A }  <->  ( -u x  e.  RR  /\  -u -u x  e.  A ) )
185, 14, 17sylanbrc 411 . . 3  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  -u x  e.  { n  e.  RR  |  -u n  e.  A } )
19 negeq 7871 . . . . . . 7  |-  ( n  =  y  ->  -u n  =  -u y )
2019eleq1d 2181 . . . . . 6  |-  ( n  =  y  ->  ( -u n  e.  A  <->  -u y  e.  A ) )
2120elrab 2807 . . . . 5  |-  ( y  e.  { n  e.  RR  |  -u n  e.  A }  <->  ( y  e.  RR  /\  -u y  e.  A ) )
22 simpr 109 . . . . . 6  |-  ( ( y  e.  RR  /\  -u y  e.  A )  ->  -u y  e.  A
)
2322a1i 9 . . . . 5  |-  ( A 
C_  RR  ->  ( ( y  e.  RR  /\  -u y  e.  A )  ->  -u y  e.  A
) )
2421, 23syl5bi 151 . . . 4  |-  ( A 
C_  RR  ->  ( y  e.  { n  e.  RR  |  -u n  e.  A }  ->  -u y  e.  A ) )
2524imp 123 . . 3  |-  ( ( A  C_  RR  /\  y  e.  { n  e.  RR  |  -u n  e.  A } )  ->  -u y  e.  A )
262, 7syl6com 35 . . . . . . . . . 10  |-  ( x  e.  A  ->  ( A  C_  RR  ->  x  e.  CC ) )
2726adantl 273 . . . . . . . . 9  |-  ( ( ( y  e.  RR  /\  -u y  e.  A
)  /\  x  e.  A )  ->  ( A  C_  RR  ->  x  e.  CC ) )
2827imp 123 . . . . . . . 8  |-  ( ( ( ( y  e.  RR  /\  -u y  e.  A )  /\  x  e.  A )  /\  A  C_  RR )  ->  x  e.  CC )
29 recn 7670 . . . . . . . . 9  |-  ( y  e.  RR  ->  y  e.  CC )
3029ad3antrrr 481 . . . . . . . 8  |-  ( ( ( ( y  e.  RR  /\  -u y  e.  A )  /\  x  e.  A )  /\  A  C_  RR )  ->  y  e.  CC )
31 negcon2 7931 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
3228, 30, 31syl2anc 406 . . . . . . 7  |-  ( ( ( ( y  e.  RR  /\  -u y  e.  A )  /\  x  e.  A )  /\  A  C_  RR )  ->  (
x  =  -u y  <->  y  =  -u x ) )
3332exp31 359 . . . . . 6  |-  ( ( y  e.  RR  /\  -u y  e.  A )  ->  ( x  e.  A  ->  ( A  C_  RR  ->  ( x  =  -u y  <->  y  =  -u x ) ) ) )
3421, 33sylbi 120 . . . . 5  |-  ( y  e.  { n  e.  RR  |  -u n  e.  A }  ->  (
x  e.  A  -> 
( A  C_  RR  ->  ( x  =  -u y 
<->  y  =  -u x
) ) ) )
3534impcom 124 . . . 4  |-  ( ( x  e.  A  /\  y  e.  { n  e.  RR  |  -u n  e.  A } )  -> 
( A  C_  RR  ->  ( x  =  -u y 
<->  y  =  -u x
) ) )
3635impcom 124 . . 3  |-  ( ( A  C_  RR  /\  (
x  e.  A  /\  y  e.  { n  e.  RR  |  -u n  e.  A } ) )  ->  ( x  = 
-u y  <->  y  =  -u x ) )
371, 18, 25, 36f1ocnv2d 5926 . 2  |-  ( A 
C_  RR  ->  ( F : A -1-1-onto-> { n  e.  RR  |  -u n  e.  A }  /\  `' F  =  ( y  e.  {
n  e.  RR  |  -u n  e.  A }  |-> 
-u y ) ) )
3837simpld 111 1  |-  ( A 
C_  RR  ->  F : A
-1-1-onto-> { n  e.  RR  |  -u n  e.  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1312    e. wcel 1461   {crab 2392    C_ wss 3035    |-> cmpt 3947   `'ccnv 4496   -1-1-onto->wf1o 5078   CCcc 7538   RRcr 7539   -ucneg 7850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-setind 4410  ax-resscn 7630  ax-1cn 7631  ax-icn 7633  ax-addcl 7634  ax-addrcl 7635  ax-mulcl 7636  ax-addcom 7638  ax-addass 7640  ax-distr 7642  ax-i2m1 7643  ax-0id 7646  ax-rnegex 7647  ax-cnre 7649
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-sub 7851  df-neg 7852
This theorem is referenced by:  negfi  10884
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