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Theorem addcom 8998
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 8795 . . . . . . . . 9  |-  1  e.  CC
21a1i 10 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  1  e.  CC )
32, 2addcld 8854 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  +  1 )  e.  CC )
4 simpl 443 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
5 simpr 447 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
63, 4, 5adddid 8859 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) ) )
74, 5addcld 8854 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
8 1p1times 8983 . . . . . . 7  |-  ( ( A  +  B )  e.  CC  ->  (
( 1  +  1 )  x.  ( A  +  B ) )  =  ( ( A  +  B )  +  ( A  +  B
) ) )
97, 8syl 15 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  1 )  x.  ( A  +  B )
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
10 1p1times 8983 . . . . . . 7  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
11 1p1times 8983 . . . . . . 7  |-  ( B  e.  CC  ->  (
( 1  +  1 )  x.  B )  =  ( B  +  B ) )
1210, 11oveqan12d 5877 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 1  +  1 )  x.  A )  +  ( ( 1  +  1 )  x.  B ) )  =  ( ( A  +  A )  +  ( B  +  B ) ) )
136, 9, 123eqtr3rd 2324 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  ( B  +  B ) )  =  ( ( A  +  B )  +  ( A  +  B ) ) )
144, 4addcld 8854 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  A
)  e.  CC )
1514, 5, 5addassd 8857 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( A  +  A )  +  ( B  +  B ) ) )
167, 4, 5addassd 8857 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B )  +  A )  +  B
)  =  ( ( A  +  B )  +  ( A  +  B ) ) )
1713, 15, 163eqtr4d 2325 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  A )  +  B )  +  B
)  =  ( ( ( A  +  B
)  +  A )  +  B ) )
1814, 5addcld 8854 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  e.  CC )
197, 4addcld 8854 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  e.  CC )
20 addcan2 8997 . . . . 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 1182 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  A )  +  B )  +  B )  =  ( ( ( A  +  B )  +  A
)  +  B )  <-> 
( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) ) )
2217, 21mpbid 201 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( ( A  +  B )  +  A ) )
234, 4, 5addassd 8857 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  A )  +  B
)  =  ( A  +  ( A  +  B ) ) )
244, 5, 4addassd 8857 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  A
)  =  ( A  +  ( B  +  A ) ) )
2522, 23, 243eqtr3d 2323 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( A  +  B ) )  =  ( A  +  ( B  +  A ) ) )
265, 4addcld 8854 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  A
)  e.  CC )
27 addcan 8996 . . 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 1182 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( A  +  B
) )  =  ( A  +  ( B  +  A ) )  <-> 
( A  +  B
)  =  ( B  +  A ) ) )
2925, 28mpbid 201 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742
This theorem is referenced by:  addcomi  9003  add12  9025  add32  9026  add42  9028  subsub23  9056  pncan2  9058  addsub  9062  addsub12  9064  addsubeq4  9066  sub32  9081  pnpcan2  9087  ppncan  9089  sub4  9092  negsubdi2  9106  ltaddsub2  9249  leaddsub2  9251  leltadd  9258  ltaddpos2  9265  addge02  9285  conjmul  9477  recp1lt1  9654  recreclt  9655  avgle1  9951  avgle2  9952  avgle  9953  nn0nnaddcl  9996  xaddcom  10565  fzen  10811  fzshftral  10869  flzadd  10951  nn0ennn  11041  seradd  11088  bernneq2  11228  hashfz  11381  revccat  11484  shftval2  11570  shftval4  11572  crim  11600  absmax  11813  climshft2  12056  summolem3  12187  binom1dif  12291  isumshft  12298  arisum  12318  mertenslem1  12340  addcos  12454  demoivreALT  12481  dvdsaddr  12568  divalglem4  12595  divalgb  12603  gcdaddm  12708  hashdvds  12843  phiprmpw  12844  pythagtriplem2  12870  mulgnndir  14589  cnaddablx  15158  cnaddabl  15159  zaddablx  15160  cncrng  16395  ioo2bl  18299  icopnfcnv  18440  uniioombllem3  18940  fta1glem1  19551  plyremlem  19684  fta1lem  19687  vieta1lem1  19690  vieta1lem2  19691  aaliou3lem2  19723  dvradcnv  19797  pserdv2  19806  reeff1olem  19822  ptolemy  19864  logcnlem4  19992  cxpsqr  20050  atandm2  20173  atandm4  20175  atanlogsublem  20211  2efiatan  20214  dvatan  20231  birthdaylem2  20247  emcllem2  20290  fsumharmonic  20305  wilthlem1  20306  wilthlem2  20307  basellem8  20325  1sgmprm  20438  perfectlem2  20469  pntibndlem1  20738  pntibndlem2  20740  pntlemd  20743  pntlemc  20744  cnaddablo  21017  addinv  21019  cdj3lem3b  23020  bpolydiflem  24789  addcomv  25655  eldioph2lem1  26839  addcomgi  27661  stoweidlem11  27760  stoweidlem13  27762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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