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Theorem addcom 9818
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 1cnd 9658 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  1  e.  CC )
21, 1addcld 9661 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  +  1 )  e.  CC )
3 simpl 458 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
4 simpr 462 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
52, 3, 4adddid 9666 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) ) )
63, 4addcld 9661 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
7 1p1times 9803 . . . . . . 7  |-  ( ( A  +  B )  e.  CC  ->  (
( 1  +  1 )  x.  ( A  +  B ) )  =  ( ( A  +  B )  +  ( A  +  B
) ) )
86, 7syl 17 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
9 1p1times 9803 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
10 1p1times 9803 . . . . . . 7  |-  ( B  e.  CC  ->  (
( 1  +  1 )  x.  B )  =  ( B  +  B ) )
119, 10oveqan12d 6324 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) )  =  ( ( A  +  A )  +  ( B  +  B ) ) )
125, 8, 113eqtr3rd 2479 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  ( B  +  B ) )  =  ( ( A  +  B )  +  ( A  +  B ) ) )
133, 3addcld 9661 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  A
)  e.  CC )
1413, 4, 4addassd 9664 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( A  +  A )  +  ( B  +  B ) ) )
156, 3, 4addassd 9664 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  A )  +  B
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
1612, 14, 153eqtr4d 2480 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( ( A  +  B
)  +  A )  +  B ) )
1713, 4addcld 9661 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  e.  CC )
186, 3addcld 9661 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  e.  CC )
19 addcan2 9817 . . . . 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 ) ) )
2017, 18, 4, 19syl3anc 1264 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  A )  +  B )  +  B )  =  ( ( ( A  +  B )  +  A
)  +  B )  <-> 
( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) ) )
2116, 20mpbid 213 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) )
223, 3, 4addassd 9664 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( A  +  ( A  +  B ) ) )
233, 4, 3addassd 9664 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  =  ( A  +  ( B  +  A ) ) )
2421, 22, 233eqtr3d 2478 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( A  +  B ) )  =  ( A  +  ( B  +  A ) ) )
254, 3addcld 9661 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  A
)  e.  CC )
26 addcan 9816 . . 3  |-  ( ( A  e.  CC  /\  ( A  +  B
)  e.  CC  /\  ( B  +  A
)  e.  CC )  ->  ( ( A  +  ( A  +  B ) )  =  ( A  +  ( B  +  A ) )  <->  ( A  +  B )  =  ( B  +  A ) ) )
273, 6, 25, 26syl3anc 1264 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( A  +  B
) )  =  ( A  +  ( B  +  A ) )  <-> 
( A  +  B
)  =  ( B  +  A ) ) )
2824, 27mpbid 213 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870  (class class class)co 6305   CCcc 9536   1c1 9539    + caddc 9541    x. cmul 9543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-ltxr 9679
This theorem is referenced by:  addcomi  9823  add12  9845  add32  9846  add42  9849  subsub23  9879  pncan2  9881  addsub  9885  addsub12  9887  addsubeq4  9889  sub32  9907  pnpcan2  9913  ppncan  9915  sub4  9918  negsubdi2  9932  ltaddsub2  10088  leaddsub2  10090  leltadd  10097  ltaddpos2  10104  addge02  10124  conjmul  10323  recp1lt1  10504  recreclt  10505  avgle1  10852  avgle2  10853  avgle  10854  nn0nnaddcl  10901  xaddcom  11531  fzen  11814  fzshftral  11880  fzo0addelr  11966  elfzoext  11968  flzadd  12056  modadd2mod  12137  nn0ennn  12189  seradd  12252  bernneq2  12396  hashfz  12594  revccat  12856  2cshwcom  12900  shftval2  13117  shftval4  13119  crim  13157  absmax  13371  climshft2  13624  summolem3  13758  binom1dif  13869  isumshft  13875  arisum  13896  mertenslem1  13918  bpolydiflem  14085  addcos  14206  demoivreALT  14233  dvdsaddr  14322  divalglem4  14352  divalgb  14360  gcdaddm  14467  hashdvds  14692  phiprmpw  14693  pythagtriplem2  14730  prmgaplem7  14990  mulgnndir  16731  cnaddablx  17441  cnaddabl  17442  zaddablx  17443  cncrng  18924  ioo2bl  21722  icopnfcnv  21866  uniioombllem3  22420  fta1glem1  22991  plyremlem  23125  fta1lem  23128  vieta1lem1  23131  vieta1lem2  23132  aaliou3lem2  23164  dvradcnv  23241  pserdv2  23250  reeff1olem  23266  ptolemy  23316  logcnlem4  23455  cxpsqrt  23513  atandm2  23668  atandm4  23670  atanlogsublem  23706  2efiatan  23709  dvatan  23726  birthdaylem2  23743  emcllem2  23787  fsumharmonic  23802  wilthlem1  23858  wilthlem2  23859  basellem8  23877  1sgmprm  23990  perfectlem2  24021  pntibndlem1  24290  pntibndlem2  24292  pntlemd  24295  pntlemc  24296  cnaddablo  25923  addinv  25925  cdj3lem3b  27928  isarchi3  28342  archiabllem2c  28350  cos2h  31639  tan2h  31640  eldioph2lem1  35310  addcomgi  36445  epoo  38219  perfectALTVlem2  38233  sgoldbaltlem2  38270  fz0addcom  38405
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