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Theorem eqreznegel 9308
Description: Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
eqreznegel  |-  ( A 
C_  ZZ  ->  { z  e.  RR  |  -u z  e.  A }  =  { z  e.  ZZ  |  -u z  e.  A } )
Distinct variable group:    z, A

Proof of Theorem eqreznegel
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ssel 3057 . . . . . . . 8  |-  ( A 
C_  ZZ  ->  ( -u w  e.  A  ->  -u w  e.  ZZ )
)
2 recn 7677 . . . . . . . . 9  |-  ( w  e.  RR  ->  w  e.  CC )
3 negid 7932 . . . . . . . . . . . 12  |-  ( w  e.  CC  ->  (
w  +  -u w
)  =  0 )
4 0z 8969 . . . . . . . . . . . 12  |-  0  e.  ZZ
53, 4syl6eqel 2205 . . . . . . . . . . 11  |-  ( w  e.  CC  ->  (
w  +  -u w
)  e.  ZZ )
65pm4.71i 386 . . . . . . . . . 10  |-  ( w  e.  CC  <->  ( w  e.  CC  /\  ( w  +  -u w )  e.  ZZ ) )
7 zrevaddcl 9008 . . . . . . . . . 10  |-  ( -u w  e.  ZZ  ->  ( ( w  e.  CC  /\  ( w  +  -u w )  e.  ZZ ) 
<->  w  e.  ZZ ) )
86, 7syl5bb 191 . . . . . . . . 9  |-  ( -u w  e.  ZZ  ->  ( w  e.  CC  <->  w  e.  ZZ ) )
92, 8syl5ib 153 . . . . . . . 8  |-  ( -u w  e.  ZZ  ->  ( w  e.  RR  ->  w  e.  ZZ ) )
101, 9syl6 33 . . . . . . 7  |-  ( A 
C_  ZZ  ->  ( -u w  e.  A  ->  ( w  e.  RR  ->  w  e.  ZZ ) ) )
1110com23 78 . . . . . 6  |-  ( A 
C_  ZZ  ->  ( w  e.  RR  ->  ( -u w  e.  A  ->  w  e.  ZZ )
) )
1211impd 252 . . . . 5  |-  ( A 
C_  ZZ  ->  ( ( w  e.  RR  /\  -u w  e.  A )  ->  w  e.  ZZ ) )
13 simpr 109 . . . . . 6  |-  ( ( w  e.  RR  /\  -u w  e.  A )  ->  -u w  e.  A
)
1413a1i 9 . . . . 5  |-  ( A 
C_  ZZ  ->  ( ( w  e.  RR  /\  -u w  e.  A )  ->  -u w  e.  A
) )
1512, 14jcad 303 . . . 4  |-  ( A 
C_  ZZ  ->  ( ( w  e.  RR  /\  -u w  e.  A )  ->  ( w  e.  ZZ  /\  -u w  e.  A ) ) )
16 zre 8962 . . . . 5  |-  ( w  e.  ZZ  ->  w  e.  RR )
1716anim1i 336 . . . 4  |-  ( ( w  e.  ZZ  /\  -u w  e.  A )  ->  ( w  e.  RR  /\  -u w  e.  A ) )
1815, 17impbid1 141 . . 3  |-  ( A 
C_  ZZ  ->  ( ( w  e.  RR  /\  -u w  e.  A )  <-> 
( w  e.  ZZ  /\  -u w  e.  A
) ) )
19 negeq 7878 . . . . 5  |-  ( z  =  w  ->  -u z  =  -u w )
2019eleq1d 2183 . . . 4  |-  ( z  =  w  ->  ( -u z  e.  A  <->  -u w  e.  A ) )
2120elrab 2809 . . 3  |-  ( w  e.  { z  e.  RR  |  -u z  e.  A }  <->  ( w  e.  RR  /\  -u w  e.  A ) )
2220elrab 2809 . . 3  |-  ( w  e.  { z  e.  ZZ  |  -u z  e.  A }  <->  ( w  e.  ZZ  /\  -u w  e.  A ) )
2318, 21, 223bitr4g 222 . 2  |-  ( A 
C_  ZZ  ->  ( w  e.  { z  e.  RR  |  -u z  e.  A }  <->  w  e.  { z  e.  ZZ  |  -u z  e.  A }
) )
2423eqrdv 2113 1  |-  ( A 
C_  ZZ  ->  { z  e.  RR  |  -u z  e.  A }  =  { z  e.  ZZ  |  -u z  e.  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   {crab 2394    C_ wss 3037  (class class class)co 5728   CCcc 7545   RRcr 7546   0cc0 7547    + caddc 7550   -ucneg 7857   ZZcz 8958
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959
This theorem is referenced by: (None)
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