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Theorem subneg 7413
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 7338 . . . 4  |-  -u B  =  ( 0  -  B )
21oveq2i 5548 . . 3  |-  ( A  -  -u B )  =  ( A  -  (
0  -  B ) )
3 0cn 7162 . . . 4  |-  0  e.  CC
4 subsub 7394 . . . 4  |-  ( ( A  e.  CC  /\  0  e.  CC  /\  B  e.  CC )  ->  ( A  -  ( 0  -  B ) )  =  ( ( A  -  0 )  +  B ) )
53, 4mp3an2 1257 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  (
0  -  B ) )  =  ( ( A  -  0 )  +  B ) )
62, 5syl5eq 2126 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  -u B
)  =  ( ( A  -  0 )  +  B ) )
7 subid1 7384 . . . 4  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
87adantr 270 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  0 )  =  A )
98oveq1d 5552 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
0 )  +  B
)  =  ( A  +  B ) )
106, 9eqtrd 2114 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  -u B
)  =  ( A  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434  (class class class)co 5537   CCcc 7030   0cc0 7032    + caddc 7035    - cmin 7335   -ucneg 7336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-setind 4282  ax-resscn 7119  ax-1cn 7120  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-addcom 7127  ax-addass 7129  ax-distr 7131  ax-i2m1 7132  ax-0id 7135  ax-rnegex 7136  ax-cnre 7138
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-sub 7337  df-neg 7338
This theorem is referenced by:  negneg  7414  negdi  7421  neg2sub  7424  subnegi  7443  subnegd  7482  recextlem1  7797  fzshftral  9190  shftval4  9843  summodnegmod  10360
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