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Theorem addid2 9011
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 9009 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
2 cnegex 9009 . . . . . 6  |-  ( x  e.  CC  ->  E. y  e.  CC  ( x  +  y )  =  0 )
32ad2antrl 708 . . . . 5  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  E. y  e.  CC  ( x  +  y )  =  0 )
4 0cn 8847 . . . . . . . . . . . 12  |-  0  e.  CC
5 addass 8840 . . . . . . . . . . . 12  |-  ( ( 0  e.  CC  /\  0  e.  CC  /\  y  e.  CC )  ->  (
( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
64, 4, 5mp3an12 1267 . . . . . . . . . . 11  |-  ( y  e.  CC  ->  (
( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
76adantr 451 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  ( x  +  y
)  =  0 )  ->  ( ( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
873ad2ant3 978 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
9 00id 9003 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
109oveq1i 5884 . . . . . . . . . 10  |-  ( ( 0  +  0 )  +  y )  =  ( 0  +  y )
11 simp1 955 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  ->  A  e.  CC )
12 simp2l 981 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  ->  x  e.  CC )
13 simp3l 983 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
y  e.  CC )
1411, 12, 13addassd 8873 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( A  +  x )  +  y )  =  ( A  +  ( x  +  y ) ) )
15 simp2r 982 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  x
)  =  0 )
1615oveq1d 5889 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( A  +  x )  +  y )  =  ( 0  +  y ) )
17 simp3r 984 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( x  +  y )  =  0 )
1817oveq2d 5890 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  ( x  +  y ) )  =  ( A  +  0 ) )
1914, 16, 183eqtr3rd 2337 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  ( 0  +  y ) )
20 addid1 9008 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
21203ad2ant1 976 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  A )
2219, 21eqtr3d 2330 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  y )  =  A )
2310, 22syl5eq 2340 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  A )
2422oveq2d 5890 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  ( 0  +  y ) )  =  ( 0  +  A ) )
258, 23, 243eqtr3rd 2337 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  A
)  =  A )
26253expia 1153 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( (
y  e.  CC  /\  ( x  +  y
)  =  0 )  ->  ( 0  +  A )  =  A ) )
2726exp3a 425 . . . . . 6  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( y  e.  CC  ->  ( (
x  +  y )  =  0  ->  (
0  +  A )  =  A ) ) )
2827rexlimdv 2679 . . . . 5  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( E. y  e.  CC  (
x  +  y )  =  0  ->  (
0  +  A )  =  A ) )
293, 28mpd 14 . . . 4  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( 0  +  A )  =  A )
3029exp32 588 . . 3  |-  ( A  e.  CC  ->  (
x  e.  CC  ->  ( ( A  +  x
)  =  0  -> 
( 0  +  A
)  =  A ) ) )
3130rexlimdv 2679 . 2  |-  ( A  e.  CC  ->  ( E. x  e.  CC  ( A  +  x
)  =  0  -> 
( 0  +  A
)  =  A ) )
321, 31mpd 14 1  |-  ( A  e.  CC  ->  (
0  +  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756
This theorem is referenced by:  addcan  9012  addid2i  9016  addid2d  9029  negneg  9113  uzindOLD  10122  fzoaddel2  10923  modid  11009  swrds1  11489  isercolllem3  12156  sumrblem  12200  summolem2a  12204  fsum0diag2  12261  eftlub  12405  gcdid  12726  cnaddablx  15174  cnaddabl  15175  cncrng  16411  ptolemy  19880  logtayl  20023  leibpilem2  20253  cnaddablo  21033  cnid  21034  axcontlem2  24665  stoweidlem1  27853  stoweidlem13  27865  stoweidlem34  27886  usgraexvlem  28261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888
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