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Theorem addsubass 9826
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 996 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 subcl 9815 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
323adant1 1014 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
4 simp3 998 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
51, 3, 4addassd 9614 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( B  -  C ) )  +  C )  =  ( A  +  ( ( B  -  C )  +  C
) ) )
6 npcan 9825 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  +  C
)  =  B )
763adant1 1014 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  +  C )  =  B )
87oveq2d 6298 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( ( B  -  C )  +  C ) )  =  ( A  +  B
) )
95, 8eqtrd 2508 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( B  -  C ) )  +  C )  =  ( A  +  B ) )
109oveq1d 6297 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  +  ( B  -  C
) )  +  C
)  -  C )  =  ( ( A  +  B )  -  C ) )
111, 3addcld 9611 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  -  C ) )  e.  CC )
12 pncan 9822 . . 3  |-  ( ( ( A  +  ( B  -  C ) )  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  +  ( B  -  C ) )  +  C )  -  C
)  =  ( A  +  ( B  -  C ) ) )
1311, 4, 12syl2anc 661 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  +  ( B  -  C
) )  +  C
)  -  C )  =  ( A  +  ( B  -  C
) ) )
1410, 13eqtr3d 2510 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  -  C )  =  ( A  +  ( B  -  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767  (class class class)co 6282   CCcc 9486    + caddc 9491    - cmin 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-ltxr 9629  df-sub 9803
This theorem is referenced by:  addsub  9827  subadd23  9828  addsubeq4  9831  npncan  9836  subsub  9845  subsub3  9847  addsub4  9858  negsub  9863  addsubassi  9906  addsubassd  9946  zeo  10942  fzen2  12043  odd2np1  13901  chtub  23215  axcontlem2  23944  numclwwlkovf2ex  24763  numclwlk2lem2f  24780  fsumcube  29399  stoweidlem26  31326  fourierdlem101  31508
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