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Theorem addid2 9752
Description:  0 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
addid2  |-  ( A  e.  CC  ->  (
0  +  A )  =  A )

Proof of Theorem addid2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnegex 9750 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
2 cnegex 9750 . . . 4  |-  ( x  e.  CC  ->  E. y  e.  CC  ( x  +  y )  =  0 )
32ad2antrl 725 . . 3  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  E. y  e.  CC  ( x  +  y )  =  0 )
4 0cn 9577 . . . . . . . . . 10  |-  0  e.  CC
5 addass 9568 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  0  e.  CC  /\  y  e.  CC )  ->  (
( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
64, 4, 5mp3an12 1312 . . . . . . . . 9  |-  ( y  e.  CC  ->  (
( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
76adantr 463 . . . . . . . 8  |-  ( ( y  e.  CC  /\  ( x  +  y
)  =  0 )  ->  ( ( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
873ad2ant3 1017 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
9 00id 9744 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
109oveq1i 6280 . . . . . . . 8  |-  ( ( 0  +  0 )  +  y )  =  ( 0  +  y )
11 simp1 994 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  ->  A  e.  CC )
12 simp2l 1020 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  ->  x  e.  CC )
13 simp3l 1022 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
y  e.  CC )
1411, 12, 13addassd 9607 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( A  +  x )  +  y )  =  ( A  +  ( x  +  y ) ) )
15 simp2r 1021 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  x
)  =  0 )
1615oveq1d 6285 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( A  +  x )  +  y )  =  ( 0  +  y ) )
17 simp3r 1023 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( x  +  y )  =  0 )
1817oveq2d 6286 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  ( x  +  y ) )  =  ( A  +  0 ) )
1914, 16, 183eqtr3rd 2504 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  ( 0  +  y ) )
20 addid1 9749 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
21203ad2ant1 1015 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  A )
2219, 21eqtr3d 2497 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  y )  =  A )
2310, 22syl5eq 2507 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  A )
2422oveq2d 6286 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  ( 0  +  y ) )  =  ( 0  +  A ) )
258, 23, 243eqtr3rd 2504 . . . . . 6  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  A
)  =  A )
26253expia 1196 . . . . 5  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( (
y  e.  CC  /\  ( x  +  y
)  =  0 )  ->  ( 0  +  A )  =  A ) )
2726expd 434 . . . 4  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( y  e.  CC  ->  ( (
x  +  y )  =  0  ->  (
0  +  A )  =  A ) ) )
2827rexlimdv 2944 . . 3  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( E. y  e.  CC  (
x  +  y )  =  0  ->  (
0  +  A )  =  A ) )
293, 28mpd 15 . 2  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( 0  +  A )  =  A )
301, 29rexlimddv 2950 1  |-  ( A  e.  CC  ->  (
0  +  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805  (class class class)co 6270   CCcc 9479   0cc0 9481    + caddc 9484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622
This theorem is referenced by:  addcan  9753  addid2i  9757  addid2d  9770  negneg  9860  uzindOLD  10953  fzo0addel  11851  fzoaddel2  11853  modid  12002  swrdspsleq  12665  swrds1  12667  isercolllem3  13571  sumrblem  13615  summolem2a  13619  fsum0diag2  13680  eftlub  13926  gcdid  14253  cnaddablx  17073  cnaddabl  17074  cncrng  18634  ptolemy  23055  logtayl  23209  leibpilem2  23469  axcontlem2  24470  usgraexvlem  24597  cnaddablo  25550  cnid  25551  dvcosax  31962  2zrngamnd  33001  aacllem  33604
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