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Theorem subneg 8241
Description: Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
subneg  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  -u B
)  =  ( A  +  B ) )

Proof of Theorem subneg
StepHypRef Expression
1 df-neg 8166 . . . 4  |-  -u B  =  ( 0  -  B )
21oveq2i 5911 . . 3  |-  ( A  -  -u B )  =  ( A  -  (
0  -  B ) )
3 0cn 7984 . . . 4  |-  0  e.  CC
4 subsub 8222 . . . 4  |-  ( ( A  e.  CC  /\  0  e.  CC  /\  B  e.  CC )  ->  ( A  -  ( 0  -  B ) )  =  ( ( A  -  0 )  +  B ) )
53, 4mp3an2 1336 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  (
0  -  B ) )  =  ( ( A  -  0 )  +  B ) )
62, 5eqtrid 2234 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  -u B
)  =  ( ( A  -  0 )  +  B ) )
7 subid1 8212 . . . 4  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
87adantr 276 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  0 )  =  A )
98oveq1d 5915 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
0 )  +  B
)  =  ( A  +  B ) )
106, 9eqtrd 2222 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  -u B
)  =  ( A  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160  (class class class)co 5900   CCcc 7844   0cc0 7846    + caddc 7849    - cmin 8163   -ucneg 8164
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 4139  ax-pow 4195  ax-pr 4230  ax-setind 4557  ax-resscn 7938  ax-1cn 7939  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-distr 7950  ax-i2m1 7951  ax-0id 7954  ax-rnegex 7955  ax-cnre 7957
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 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-sub 8165  df-neg 8166
This theorem is referenced by:  negneg  8242  negdi  8249  neg2sub  8252  subnegi  8271  subnegd  8310  recextlem1  8643  fzshftral  10144  shftval4  10878  fsumshftm  11494  eftlub  11739  summodnegmod  11870  wilthlem1  14883
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