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Theorem addsubass 7284
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 915 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 subcl 7273 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
323adant1 933 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
4 simp3 917 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
51, 3, 4addassd 7107 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( B  -  C ) )  +  C )  =  ( A  +  ( ( B  -  C )  +  C
) ) )
6 npcan 7283 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  +  C
)  =  B )
763adant1 933 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  +  C )  =  B )
87oveq2d 5556 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( ( B  -  C )  +  C ) )  =  ( A  +  B
) )
95, 8eqtrd 2088 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( B  -  C ) )  +  C )  =  ( A  +  B ) )
109oveq1d 5555 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  +  ( B  -  C
) )  +  C
)  -  C )  =  ( ( A  +  B )  -  C ) )
111, 3addcld 7104 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  -  C ) )  e.  CC )
12 pncan 7280 . . 3  |-  ( ( ( A  +  ( B  -  C ) )  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  +  ( B  -  C ) )  +  C )  -  C
)  =  ( A  +  ( B  -  C ) ) )
1311, 4, 12syl2anc 397 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  +  ( B  -  C
) )  +  C
)  -  C )  =  ( A  +  ( B  -  C
) ) )
1410, 13eqtr3d 2090 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 896    = wceq 1259    e. wcel 1409  (class class class)co 5540   CCcc 6945    + caddc 6950    - cmin 7245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247
This theorem is referenced by:  addsub  7285  subadd23  7286  addsubeq4  7289  npncan  7295  subsub  7304  subsub3  7306  addsub4  7317  negsub  7322  addsubassi  7365  addsubassd  7405  zeo  8402  frecfzen2  9368  odd2np1  10184
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