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Theorem addcom 8878
Description: Addition commutes. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
addcom  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )

Proof of Theorem addcom
StepHypRef Expression
1 ax-1cn 8675 . . . . . . . . 9  |-  1  e.  CC
21a1i 12 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  1  e.  CC )
32, 2addcld 8734 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  +  1 )  e.  CC )
4 simpl 445 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
5 simpr 449 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
63, 4, 5adddid 8739 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) ) )
74, 5addcld 8734 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 1p1times 8863 . . . . . . 7  |-  ( ( A  +  B )  e.  CC  ->  (
( 1  +  1 )  x.  ( A  +  B ) )  =  ( ( A  +  B )  +  ( A  +  B
) ) )
97, 8syl 17 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
10 1p1times 8863 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
11 1p1times 8863 . . . . . . 7  |-  ( B  e.  CC  ->  (
( 1  +  1 )  x.  B )  =  ( B  +  B ) )
1210, 11oveqan12d 5729 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) )  =  ( ( A  +  A )  +  ( B  +  B ) ) )
136, 9, 123eqtr3rd 2294 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  ( B  +  B ) )  =  ( ( A  +  B )  +  ( A  +  B ) ) )
144, 4addcld 8734 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  A
)  e.  CC )
1514, 5, 5addassd 8737 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( A  +  A )  +  ( B  +  B ) ) )
167, 4, 5addassd 8737 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  A )  +  B
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
1713, 15, 163eqtr4d 2295 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( ( A  +  B
)  +  A )  +  B ) )
1814, 5addcld 8734 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  e.  CC )
197, 4addcld 8734 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  e.  CC )
20 addcan2 8877 . . . . 5  |-  ( ( ( ( A  +  A )  +  B
)  e.  CC  /\  ( ( A  +  B )  +  A
)  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  A )  +  B )  +  B )  =  ( ( ( A  +  B )  +  A
)  +  B )  <-> 
( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) ) )
2118, 19, 5, 20syl3anc 1187 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  A )  +  B )  +  B )  =  ( ( ( A  +  B )  +  A
)  +  B )  <-> 
( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) ) )
2217, 21mpbid 203 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) )
234, 4, 5addassd 8737 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( A  +  ( A  +  B ) ) )
244, 5, 4addassd 8737 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  =  ( A  +  ( B  +  A ) ) )
2522, 23, 243eqtr3d 2293 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( A  +  B ) )  =  ( A  +  ( B  +  A ) ) )
265, 4addcld 8734 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  A
)  e.  CC )
27 addcan 8876 . . 3  |-  ( ( A  e.  CC  /\  ( A  +  B
)  e.  CC  /\  ( B  +  A
)  e.  CC )  ->  ( ( A  +  ( A  +  B ) )  =  ( A  +  ( B  +  A ) )  <->  ( A  +  B )  =  ( B  +  A ) ) )
284, 7, 26, 27syl3anc 1187 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( A  +  B
) )  =  ( A  +  ( B  +  A ) )  <-> 
( A  +  B
)  =  ( B  +  A ) ) )
2925, 28mpbid 203 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621  (class class class)co 5710   CCcc 8615   1c1 8618    + caddc 8620    x. cmul 8622
This theorem is referenced by:  addcomi  8883  add12  8905  add32  8906  add42  8908  subsub23  8936  pncan2  8938  addsub  8942  addsub12  8944  addsubeq4  8946  sub32  8961  pnpcan2  8967  ppncan  8969  sub4  8972  negsubdi2  8986  ltaddsub2  9129  leaddsub2  9131  leltadd  9138  ltaddpos2  9145  addge02  9165  conjmul  9357  recp1lt1  9534  recreclt  9535  avgle1  9830  avgle2  9831  avgle  9832  nn0nnaddcl  9875  xaddcom  10443  fzen  10689  fzshftral  10747  flzadd  10829  nn0ennn  10919  seradd  10966  bernneq2  11106  hashfz  11258  revccat  11361  shftval2  11447  shftval4  11449  crim  11477  absmax  11690  climshft2  11933  summolem3  12064  binom1dif  12168  isumshft  12172  arisum  12192  mertenslem1  12214  addcos  12328  demoivreALT  12355  dvdsaddr  12442  divalglem4  12469  divalgb  12477  gcdaddm  12582  hashdvds  12717  phiprmpw  12718  pythagtriplem2  12744  mulgnndir  14424  cnaddablx  14993  cnaddabl  14994  zaddablx  14995  cncrng  16227  ioo2bl  18131  icopnfcnv  18272  uniioombllem3  18772  fta1glem1  19383  plyremlem  19516  fta1lem  19519  vieta1lem1  19522  vieta1lem2  19523  aaliou3lem2  19555  dvradcnv  19629  pserdv2  19638  reeff1olem  19654  ptolemy  19696  logcnlem4  19824  cxpsqr  19882  atandm2  20005  atandm4  20007  atanlogsublem  20043  2efiatan  20046  dvatan  20063  birthdaylem2  20079  emcllem2  20122  fsumharmonic  20137  wilthlem1  20138  wilthlem2  20139  basellem8  20157  1sgmprm  20270  perfectlem2  20301  pntibndlem1  20570  pntibndlem2  20572  pntlemd  20575  pntlemc  20576  cnaddablo  20847  addinv  20849  cdj3lem3b  22850  bpolydiflem  23963  addcomv  24821  eldioph2lem1  26005  addcomgi  26828  stoweidlem11  26894  stoweidlem13  26896
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752
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