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Theorem addcom 9208
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 9004 . . . . . . . . 9  |-  1  e.  CC
21a1i 11 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  1  e.  CC )
32, 2addcld 9063 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  +  1 )  e.  CC )
4 simpl 444 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
5 simpr 448 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
63, 4, 5adddid 9068 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) ) )
74, 5addcld 9063 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 1p1times 9193 . . . . . . 7  |-  ( ( A  +  B )  e.  CC  ->  (
( 1  +  1 )  x.  ( A  +  B ) )  =  ( ( A  +  B )  +  ( A  +  B
) ) )
97, 8syl 16 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
10 1p1times 9193 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
11 1p1times 9193 . . . . . . 7  |-  ( B  e.  CC  ->  (
( 1  +  1 )  x.  B )  =  ( B  +  B ) )
1210, 11oveqan12d 6059 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) )  =  ( ( A  +  A )  +  ( B  +  B ) ) )
136, 9, 123eqtr3rd 2445 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  ( B  +  B ) )  =  ( ( A  +  B )  +  ( A  +  B ) ) )
144, 4addcld 9063 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  A
)  e.  CC )
1514, 5, 5addassd 9066 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( A  +  A )  +  ( B  +  B ) ) )
167, 4, 5addassd 9066 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  A )  +  B
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
1713, 15, 163eqtr4d 2446 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( ( A  +  B
)  +  A )  +  B ) )
1814, 5addcld 9063 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  e.  CC )
197, 4addcld 9063 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  e.  CC )
20 addcan2 9207 . . . . 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 1184 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  A )  +  B )  +  B )  =  ( ( ( A  +  B )  +  A
)  +  B )  <-> 
( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) ) )
2217, 21mpbid 202 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) )
234, 4, 5addassd 9066 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( A  +  ( A  +  B ) ) )
244, 5, 4addassd 9066 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  =  ( A  +  ( B  +  A ) ) )
2522, 23, 243eqtr3d 2444 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( A  +  B ) )  =  ( A  +  ( B  +  A ) ) )
265, 4addcld 9063 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  A
)  e.  CC )
27 addcan 9206 . . 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 1184 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( A  +  B
) )  =  ( A  +  ( B  +  A ) )  <-> 
( A  +  B
)  =  ( B  +  A ) ) )
2925, 28mpbid 202 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721  (class class class)co 6040   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951
This theorem is referenced by:  addcomi  9213  add12  9235  add32  9236  add42  9238  subsub23  9266  pncan2  9268  addsub  9272  addsub12  9274  addsubeq4  9276  sub32  9291  pnpcan2  9297  ppncan  9299  sub4  9302  negsubdi2  9316  ltaddsub2  9459  leaddsub2  9461  leltadd  9468  ltaddpos2  9475  addge02  9495  conjmul  9687  recp1lt1  9864  recreclt  9865  avgle1  10163  avgle2  10164  avgle  10165  nn0nnaddcl  10208  xaddcom  10780  fzen  11028  fzshftral  11089  flzadd  11183  nn0ennn  11273  seradd  11320  bernneq2  11461  hashfz  11647  revccat  11753  shftval2  11845  shftval4  11847  crim  11875  absmax  12088  climshft2  12331  summolem3  12463  binom1dif  12567  isumshft  12574  arisum  12594  mertenslem1  12616  addcos  12730  demoivreALT  12757  dvdsaddr  12844  divalglem4  12871  divalgb  12879  gcdaddm  12984  hashdvds  13119  phiprmpw  13120  pythagtriplem2  13146  mulgnndir  14867  cnaddablx  15436  cnaddabl  15437  zaddablx  15438  cncrng  16677  ioo2bl  18777  icopnfcnv  18920  uniioombllem3  19430  fta1glem1  20041  plyremlem  20174  fta1lem  20177  vieta1lem1  20180  vieta1lem2  20181  aaliou3lem2  20213  dvradcnv  20290  pserdv2  20299  reeff1olem  20315  ptolemy  20357  logcnlem4  20489  cxpsqr  20547  atandm2  20670  atandm4  20672  atanlogsublem  20708  2efiatan  20711  dvatan  20728  birthdaylem2  20744  emcllem2  20788  fsumharmonic  20803  wilthlem1  20804  wilthlem2  20805  basellem8  20823  1sgmprm  20936  perfectlem2  20967  pntibndlem1  21236  pntibndlem2  21238  pntlemd  21241  pntlemc  21242  cnaddablo  21891  addinv  21893  cdj3lem3b  23896  bpolydiflem  26004  eldioph2lem1  26708  addcomgi  27528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081
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