MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addsubass Structured version   Unicode version

Theorem addsubass 9625
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 988 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 subcl 9614 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
323adant1 1006 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
4 simp3 990 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
51, 3, 4addassd 9413 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( B  -  C ) )  +  C )  =  ( A  +  ( ( B  -  C )  +  C
) ) )
6 npcan 9624 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  +  C
)  =  B )
763adant1 1006 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  C
)  +  C )  =  B )
87oveq2d 6112 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( ( B  -  C )  +  C ) )  =  ( A  +  B
) )
95, 8eqtrd 2475 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( B  -  C ) )  +  C )  =  ( A  +  B ) )
109oveq1d 6111 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  +  ( B  -  C
) )  +  C
)  -  C )  =  ( ( A  +  B )  -  C ) )
111, 3addcld 9410 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  -  C ) )  e.  CC )
12 pncan 9621 . . 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 2477 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 965    = wceq 1369    e. wcel 1756  (class class class)co 6096   CCcc 9285    + caddc 9290    - cmin 9600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-ltxr 9428  df-sub 9602
This theorem is referenced by:  addsub  9626  subadd23  9627  addsubeq4  9630  npncan  9635  subsub  9644  subsub3  9646  addsub4  9657  negsub  9662  addsubassi  9704  addsubassd  9744  zeo  10732  fzen2  11796  odd2np1  13597  chtub  22556  axcontlem2  23216  fsumcube  28208  stoweidlem26  29826  numclwwlkovf2ex  30684  numclwlk2lem2f  30701
  Copyright terms: Public domain W3C validator