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Theorem addid0 8128
Description: If adding a number to a another number yields the other number, the added number must be  0. This shows that  0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)
Assertion
Ref Expression
addid0  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  +  Y )  =  X  <-> 
Y  =  0 ) )

Proof of Theorem addid0
StepHypRef Expression
1 simpl 108 . . . 4  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  X  e.  CC )
2 simpr 109 . . . 4  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  Y  e.  CC )
31, 1, 2subaddd 8084 . . 3  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  -  X )  =  Y  <-> 
( X  +  Y
)  =  X ) )
4 eqcom 2139 . . . . 5  |-  ( ( X  -  X )  =  Y  <->  Y  =  ( X  -  X
) )
5 simpr 109 . . . . . . 7  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  ->  Y  =  ( X  -  X ) )
6 subid 7974 . . . . . . . 8  |-  ( X  e.  CC  ->  ( X  -  X )  =  0 )
76adantr 274 . . . . . . 7  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  -> 
( X  -  X
)  =  0 )
85, 7eqtrd 2170 . . . . . 6  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  ->  Y  =  0 )
98ex 114 . . . . 5  |-  ( X  e.  CC  ->  ( Y  =  ( X  -  X )  ->  Y  =  0 ) )
104, 9syl5bi 151 . . . 4  |-  ( X  e.  CC  ->  (
( X  -  X
)  =  Y  ->  Y  =  0 ) )
1110adantr 274 . . 3  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  -  X )  =  Y  ->  Y  =  0 ) )
123, 11sylbird 169 . 2  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  +  Y )  =  X  ->  Y  =  0 ) )
13 oveq2 5775 . . . . 5  |-  ( Y  =  0  ->  ( X  +  Y )  =  ( X  + 
0 ) )
14 addid1 7893 . . . . 5  |-  ( X  e.  CC  ->  ( X  +  0 )  =  X )
1513, 14sylan9eqr 2192 . . . 4  |-  ( ( X  e.  CC  /\  Y  =  0 )  ->  ( X  +  Y )  =  X )
1615ex 114 . . 3  |-  ( X  e.  CC  ->  ( Y  =  0  ->  ( X  +  Y )  =  X ) )
1716adantr 274 . 2  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( Y  =  0  ->  ( X  +  Y )  =  X ) )
1812, 17impbid 128 1  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  +  Y )  =  X  <-> 
Y  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480  (class class class)co 5767   CCcc 7611   0cc0 7613    + caddc 7616    - cmin 7926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-sub 7928
This theorem is referenced by:  addn0nid  8129
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