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Theorem addsubass 7692
Description: Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
addsubass  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  C )  =  ( A  +  ( B  -  C
) ) )

Proof of Theorem addsubass
StepHypRef Expression
1 simp1 943 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 subcl 7681 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
323adant1 961 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
4 simp3 945 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
51, 3, 4addassd 7510 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( B  -  C ) )  +  C )  =  ( A  +  ( ( B  -  C )  +  C
) ) )
6 npcan 7691 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  +  C
)  =  B )
763adant1 961 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  +  C )  =  B )
87oveq2d 5668 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( ( B  -  C )  +  C ) )  =  ( A  +  B
) )
95, 8eqtrd 2120 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( B  -  C ) )  +  C )  =  ( A  +  B ) )
109oveq1d 5667 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  +  ( B  -  C
) )  +  C
)  -  C )  =  ( ( A  +  B )  -  C ) )
111, 3addcld 7507 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  -  C ) )  e.  CC )
12 pncan 7688 . . 3  |-  ( ( ( A  +  ( B  -  C ) )  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  +  ( B  -  C ) )  +  C )  -  C
)  =  ( A  +  ( B  -  C ) ) )
1311, 4, 12syl2anc 403 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  +  ( B  -  C
) )  +  C
)  -  C )  =  ( A  +  ( B  -  C
) ) )
1410, 13eqtr3d 2122 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  C )  =  ( A  +  ( B  -  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 924    = wceq 1289    e. wcel 1438  (class class class)co 5652   CCcc 7348    + caddc 7353    - cmin 7653
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7655
This theorem is referenced by:  addsub  7693  subadd23  7694  addsubeq4  7697  npncan  7703  subsub  7712  subsub3  7714  addsub4  7725  negsub  7730  addsubassi  7773  addsubassd  7813  zeo  8851  frecfzen2  9834  odd2np1  11151
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