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Theorem addid2 8995
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 8993 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
2 cnegex 8993 . . . . . 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 8831 . . . . . . . . . . . 12  |-  0  e.  CC
5 addass 8824 . . . . . . . . . . . 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 8987 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
109oveq1i 5868 . . . . . . . . . 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 8857 . . . . . . . . . . . 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 5873 . . . . . . . . . . . 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 5874 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  ( x  +  y ) )  =  ( A  +  0 ) )
1914, 16, 183eqtr3rd 2324 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  ( 0  +  y ) )
20 addid1 8992 . . . . . . . . . . . 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 2317 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  y )  =  A )
2310, 22syl5eq 2327 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  A )
2422oveq2d 5874 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  ( 0  +  y ) )  =  ( 0  +  A ) )
258, 23, 243eqtr3rd 2324 . . . . . . . 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 2666 . . . . 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 2666 . 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 1623    e. wcel 1684   E.wrex 2544  (class class class)co 5858   CCcc 8735   0cc0 8737    + caddc 8740
This theorem is referenced by:  addcan  8996  addid2i  9000  addid2d  9013  negneg  9097  uzindOLD  10106  fzoaddel2  10907  modid  10993  swrds1  11473  isercolllem3  12140  sumrblem  12184  summolem2a  12188  fsum0diag2  12245  eftlub  12389  gcdid  12710  cnaddablx  15158  cnaddabl  15159  cncrng  16395  ptolemy  19864  logtayl  20007  leibpilem2  20237  cnaddablo  21017  cnid  21018  axcontlem2  24593  stoweidlem1  27750  stoweidlem13  27762  stoweidlem34  27783  usgraexvlem  28127
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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