MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addid2 Structured version   Unicode version

Theorem addid2 9249
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 9247 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
2 cnegex 9247 . . . 4  |-  ( x  e.  CC  ->  E. y  e.  CC  ( x  +  y )  =  0 )
32ad2antrl 709 . . 3  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  E. y  e.  CC  ( x  +  y )  =  0 )
4 0cn 9084 . . . . . . . . . 10  |-  0  e.  CC
5 addass 9077 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  0  e.  CC  /\  y  e.  CC )  ->  (
( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
64, 4, 5mp3an12 1269 . . . . . . . . 9  |-  ( y  e.  CC  ->  (
( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
76adantr 452 . . . . . . . 8  |-  ( ( y  e.  CC  /\  ( x  +  y
)  =  0 )  ->  ( ( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
873ad2ant3 980 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  ( 0  +  ( 0  +  y ) ) )
9 00id 9241 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
109oveq1i 6091 . . . . . . . 8  |-  ( ( 0  +  0 )  +  y )  =  ( 0  +  y )
11 simp1 957 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  ->  A  e.  CC )
12 simp2l 983 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  ->  x  e.  CC )
13 simp3l 985 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
y  e.  CC )
1411, 12, 13addassd 9110 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( A  +  x )  +  y )  =  ( A  +  ( x  +  y ) ) )
15 simp2r 984 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  x
)  =  0 )
1615oveq1d 6096 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( A  +  x )  +  y )  =  ( 0  +  y ) )
17 simp3r 986 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( x  +  y )  =  0 )
1817oveq2d 6097 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  ( x  +  y ) )  =  ( A  +  0 ) )
1914, 16, 183eqtr3rd 2477 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  ( 0  +  y ) )
20 addid1 9246 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
21203ad2ant1 978 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( A  +  0 )  =  A )
2219, 21eqtr3d 2470 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  y )  =  A )
2310, 22syl5eq 2480 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( ( 0  +  0 )  +  y )  =  A )
2422oveq2d 6097 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  ( 0  +  y ) )  =  ( 0  +  A ) )
258, 23, 243eqtr3rd 2477 . . . . . 6  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 )  /\  ( y  e.  CC  /\  ( x  +  y )  =  0 ) )  -> 
( 0  +  A
)  =  A )
26253expia 1155 . . . . 5  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( (
y  e.  CC  /\  ( x  +  y
)  =  0 )  ->  ( 0  +  A )  =  A ) )
2726exp3a 426 . . . 4  |-  ( ( A  e.  CC  /\  ( x  e.  CC  /\  ( A  +  x
)  =  0 ) )  ->  ( y  e.  CC  ->  ( (
x  +  y )  =  0  ->  (
0  +  A )  =  A ) ) )
2827rexlimdv 2829 . . 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 2834 1  |-  ( A  e.  CC  ->  (
0  +  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706  (class class class)co 6081   CCcc 8988   0cc0 8990    + caddc 8993
This theorem is referenced by:  addcan  9250  addid2i  9254  addid2d  9267  negneg  9351  uzindOLD  10364  fzoaddel2  11176  modid  11270  swrds1  11787  isercolllem3  12460  sumrblem  12505  summolem2a  12509  fsum0diag2  12566  eftlub  12710  gcdid  13031  cnaddablx  15481  cnaddabl  15482  cncrng  16722  ptolemy  20404  logtayl  20551  leibpilem2  20781  usgraexvlem  21414  cnaddablo  21938  cnid  21939  axcontlem2  25904  swrd0swrdid  28200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125
  Copyright terms: Public domain W3C validator