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Theorem addid2 9551
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 9549 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
2 cnegex 9549 . . . 4  |-  ( x  e.  CC  ->  E. y  e.  CC  ( x  +  y )  =  0 )
32ad2antrl 727 . . 3  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  E. y  e.  CC  ( x  +  y )  =  0 )
4 0cn 9377 . . . . . . . . . 10  |-  0  e.  CC
5 addass 9368 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  0  e.  CC  /\  y  e.  CC )  ->  (
( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
64, 4, 5mp3an12 1304 . . . . . . . . 9  |-  ( y  e.  CC  ->  (
( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
76adantr 465 . . . . . . . 8  |-  ( ( y  e.  CC  /\  ( x  +  y
)  =  0 )  ->  ( ( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
873ad2ant3 1011 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
9 00id 9543 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
109oveq1i 6100 . . . . . . . 8  |-  ( ( 0  +  0 )  +  y )  =  ( 0  +  y )
11 simp1 988 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  ->  A  e.  CC )
12 simp2l 1014 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  ->  x  e.  CC )
13 simp3l 1016 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
y  e.  CC )
1411, 12, 13addassd 9407 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( A  +  x )  +  y )  =  ( A  +  ( x  +  y ) ) )
15 simp2r 1015 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  x
)  =  0 )
1615oveq1d 6105 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( A  +  x )  +  y )  =  ( 0  +  y ) )
17 simp3r 1017 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( x  +  y )  =  0 )
1817oveq2d 6106 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  ( x  +  y ) )  =  ( A  +  0 ) )
1914, 16, 183eqtr3rd 2483 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  ( 0  +  y ) )
20 addid1 9548 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
21203ad2ant1 1009 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  A )
2219, 21eqtr3d 2476 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  y )  =  A )
2310, 22syl5eq 2486 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  A )
2422oveq2d 6106 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  ( 0  +  y ) )  =  ( 0  +  A ) )
258, 23, 243eqtr3rd 2483 . . . . . 6  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  A
)  =  A )
26253expia 1189 . . . . 5  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( (
y  e.  CC  /\  ( x  +  y
)  =  0 )  ->  ( 0  +  A )  =  A ) )
2726expd 436 . . . 4  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( y  e.  CC  ->  ( (
x  +  y )  =  0  ->  (
0  +  A )  =  A ) ) )
2827rexlimdv 2839 . . 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 2844 1  |-  ( A  e.  CC  ->  (
0  +  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2715  (class class class)co 6090   CCcc 9279   0cc0 9281    + caddc 9284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-ltxr 9422
This theorem is referenced by:  addcan  9552  addid2i  9556  addid2d  9569  negneg  9658  uzindOLD  10735  fzoaddel2  11597  modid  11731  swrds1  12344  isercolllem3  13143  sumrblem  13187  summolem2a  13191  fsum0diag2  13249  eftlub  13392  gcdid  13714  cnaddablx  16347  cnaddabl  16348  cncrng  17836  ptolemy  21957  logtayl  22104  leibpilem2  22335  axcontlem2  23210  usgraexvlem  23312  cnaddablo  23836  cnid  23837
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