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Theorem neg2sub 7986
Description: Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
Assertion
Ref Expression
neg2sub  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  ( B  -  A ) )

Proof of Theorem neg2sub
StepHypRef Expression
1 negcl 7926 . . 3  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 subneg 7975 . . 3  |-  ( (
-u A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  ( -u A  +  B ) )
31, 2sylan 279 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  (
-u A  +  B
) )
4 negsubdi 7982 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  (
-u A  +  B
) )
5 negsubdi2 7985 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  ( B  -  A ) )
63, 4, 53eqtr2d 2154 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  ( B  -  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463  (class class class)co 5740   CCcc 7582    + caddc 7587    - cmin 7897   -ucneg 7898
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 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-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420  ax-resscn 7676  ax-1cn 7677  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695
This theorem depends on definitions:  df-bi 116  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 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-sub 7899  df-neg 7900
This theorem is referenced by:  neg2subd  8054
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