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Theorem neg2sub 8135
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 8075 . . 3  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 subneg 8124 . . 3  |-  ( (
-u A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  ( -u A  +  B ) )
31, 2sylan 281 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  (
-u A  +  B
) )
4 negsubdi 8131 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  (
-u A  +  B
) )
5 negsubdi2 8134 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  ( B  -  A ) )
63, 4, 53eqtr2d 2196 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 1335    e. wcel 2128  (class class class)co 5824   CCcc 7730    + caddc 7735    - cmin 8046   -ucneg 8047
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4496  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-sub 8048  df-neg 8049
This theorem is referenced by:  neg2subd  8203
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