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Theorem negf1o 8112
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 3061 . . . . . 6  |-  ( A 
C_  RR  ->  ( x  e.  A  ->  x  e.  RR ) )
3 renegcl 7991 . . . . . 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 7721 . . . . . . . . 9  |-  ( x  e.  RR  ->  x  e.  CC )
8 negneg 7980 . . . . . . . . . 10  |-  ( x  e.  CC  ->  -u -u x  =  x )
98eqcomd 2123 . . . . . . . . 9  |-  ( x  e.  CC  ->  x  =  -u -u x )
107, 9syl 14 . . . . . . . 8  |-  ( x  e.  RR  ->  x  =  -u -u x )
1110eleq1d 2186 . . . . . . 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 275 . . . . 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 7923 . . . . . 6  |-  ( n  =  -u x  ->  -u n  =  -u -u x )
1615eleq1d 2186 . . . . 5  |-  ( n  =  -u x  ->  ( -u n  e.  A  <->  -u -u x  e.  A ) )
1716elrab 2813 . . . 4  |-  ( -u x  e.  { n  e.  RR  |  -u n  e.  A }  <->  ( -u x  e.  RR  /\  -u -u x  e.  A ) )
185, 14, 17sylanbrc 413 . . 3  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  -u x  e.  { n  e.  RR  |  -u n  e.  A } )
19 negeq 7923 . . . . . . 7  |-  ( n  =  y  ->  -u n  =  -u y )
2019eleq1d 2186 . . . . . 6  |-  ( n  =  y  ->  ( -u n  e.  A  <->  -u y  e.  A ) )
2120elrab 2813 . . . . 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 275 . . . . . . . . 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 7721 . . . . . . . . 9  |-  ( y  e.  RR  ->  y  e.  CC )
3029ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( y  e.  RR  /\  -u y  e.  A )  /\  x  e.  A )  /\  A  C_  RR )  ->  y  e.  CC )
31 negcon2 7983 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
3228, 30, 31syl2anc 408 . . . . . . 7  |-  ( ( ( ( y  e.  RR  /\  -u y  e.  A )  /\  x  e.  A )  /\  A  C_  RR )  ->  (
x  =  -u y  <->  y  =  -u x ) )
3332exp31 361 . . . . . 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 5942 . 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 1316    e. wcel 1465   {crab 2397    C_ wss 3041    |-> cmpt 3959   `'ccnv 4508   -1-1-onto->wf1o 5092   CCcc 7586   RRcr 7587   -ucneg 7902
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-neg 7904
This theorem is referenced by:  negfi  10954
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