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Theorem addcom 8874
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 8672 . . . . . . . . 9  |-  1  e.  CC
21a1i 12 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  1  e.  CC )
32, 2addcld 8731 . . . . . . 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 8736 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) ) )
74, 5addcld 8731 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 1p1times 8859 . . . . . . 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 8859 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
11 1p1times 8859 . . . . . . 7  |-  ( B  e.  CC  ->  (
( 1  +  1 )  x.  B )  =  ( B  +  B ) )
1210, 11oveqan12d 5726 . . . . . 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 8731 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  A
)  e.  CC )
1514, 5, 5addassd 8734 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( A  +  A )  +  ( B  +  B ) ) )
167, 4, 5addassd 8734 . . . . 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 8731 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  e.  CC )
197, 4addcld 8731 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  e.  CC )
20 addcan2 8873 . . . . 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 8734 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( A  +  ( A  +  B ) ) )
244, 5, 4addassd 8734 . . 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 8731 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  A
)  e.  CC )
27 addcan 8872 . . 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 5707   CCcc 8612   1c1 8615    + caddc 8617    x. cmul 8619
This theorem is referenced by:  addcomi  8879  add12  8897  add32  8898  add42  8900  subsub23  8928  pncan2  8930  addsub  8934  addsub12  8936  addsubeq4  8938  sub32  8953  pnpcan2  8959  ppncan  8961  sub4  8964  negsubdi2  8978  ltaddsub2  9110  leaddsub2  9112  leltadd  9119  ltaddpos2  9126  addge02  9146  conjmul  9333  recp1lt1  9502  recreclt  9503  avgle1  9798  avgle2  9799  avgle  9800  nn0nnaddcl  9842  xaddcom  10410  fzen  10654  fzshftral  10712  flzadd  10794  nn0ennn  10884  seradd  10931  bernneq2  11070  hashfz  11222  revccat  11325  shftval2  11411  shftval4  11413  crim  11441  absmax  11652  climshft2  11895  summolem3  12026  binom1dif  12130  isumshft  12134  arisum  12154  mertenslem1  12176  addcos  12290  demoivreALT  12317  dvdsaddr  12404  divalglem4  12431  divalgb  12439  gcdaddm  12544  hashdvds  12679  phiprmpw  12680  pythagtriplem2  12706  mulgnndir  14386  cnaddablx  14955  cnaddabl  14956  zaddablx  14957  cncrng  16189  ioo2bl  18093  icopnfcnv  18234  uniioombllem3  18734  fta1glem1  19345  plyremlem  19478  fta1lem  19481  vieta1lem1  19484  vieta1lem2  19485  aaliou3lem2  19517  dvradcnv  19591  pserdv2  19600  reeff1olem  19616  ptolemy  19656  logcnlem4  19776  cxpsqr  19852  atandm2  19939  atandm4  19941  atanlogsublem  19977  2efiatan  19980  dvatan  19997  birthdaylem2  20013  emcllem2  20056  fsumharmonic  20071  wilthlem1  20072  wilthlem2  20073  basellem8  20091  1sgmprm  20204  perfectlem2  20235  pntibndlem1  20504  pntibndlem2  20506  pntlemd  20509  pntlemc  20510  cnaddablo  20776  addinv  20778  cdj3lem3b  22779  bpolydiflem  23892  addcomv  24750  eldioph2lem1  25934  addcomgi  26757
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 4035  ax-nul 4043  ax-pow 4079  ax-pr 4105  ax-un 4400  ax-resscn 8671  ax-1cn 8672  ax-icn 8673  ax-addcl 8674  ax-addrcl 8675  ax-mulcl 8676  ax-mulrcl 8677  ax-mulcom 8678  ax-addass 8679  ax-mulass 8680  ax-distr 8681  ax-i2m1 8682  ax-1ne0 8683  ax-1rid 8684  ax-rnegex 8685  ax-rrecex 8686  ax-cnre 8687  ax-pre-lttri 8688  ax-pre-lttrn 8689  ax-pre-ltadd 8690
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 2511  df-rex 2512  df-rab 2514  df-v 2727  df-sbc 2920  df-csb 3007  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-nul 3360  df-if 3468  df-pw 3529  df-sn 3547  df-pr 3548  df-op 3550  df-uni 3725  df-br 3918  df-opab 3972  df-mpt 3973  df-id 4199  df-po 4204  df-so 4205  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 4599  df-fn 4600  df-f 4601  df-f1 4602  df-fo 4603  df-f1o 4604  df-fv 4605  df-ov 5710  df-er 6543  df-en 6747  df-dom 6748  df-sdom 6749  df-pnf 8746  df-mnf 8747  df-ltxr 8749
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