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Theorem negsubdi 8231
Description: Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
negsubdi  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  (
-u A  +  B
) )

Proof of Theorem negsubdi
StepHypRef Expression
1 0cn 7967 . . 3  |-  0  e.  CC
2 subsub 8205 . . 3  |-  ( ( 0  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
0  -  ( A  -  B ) )  =  ( ( 0  -  A )  +  B ) )
31, 2mp3an1 1335 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 0  -  ( A  -  B )
)  =  ( ( 0  -  A )  +  B ) )
4 df-neg 8149 . 2  |-  -u ( A  -  B )  =  ( 0  -  ( A  -  B
) )
5 df-neg 8149 . . 3  |-  -u A  =  ( 0  -  A )
65oveq1i 5901 . 2  |-  ( -u A  +  B )  =  ( ( 0  -  A )  +  B )
73, 4, 63eqtr4g 2247 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  (
-u A  +  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160  (class class class)co 5891   CCcc 7827   0cc0 7829    + caddc 7832    - cmin 8146   -ucneg 8147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-setind 4551  ax-resscn 7921  ax-1cn 7922  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-sub 8148  df-neg 8149
This theorem is referenced by:  negdi  8232  negsubdi2  8234  neg2sub  8235  negsubdid  8301  odd2np1  11896  sin2pim  14631  cos2pim  14632
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