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Theorem addid0 7596
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 107 . . . 4  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  X  e.  CC )
2 simpr 108 . . . 4  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  Y  e.  CC )
31, 1, 2subaddd 7556 . . 3  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  -  X )  =  Y  <-> 
( X  +  Y
)  =  X ) )
4 eqcom 2085 . . . . 5  |-  ( ( X  -  X )  =  Y  <->  Y  =  ( X  -  X
) )
5 simpr 108 . . . . . . 7  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  ->  Y  =  ( X  -  X ) )
6 subid 7446 . . . . . . . 8  |-  ( X  e.  CC  ->  ( X  -  X )  =  0 )
76adantr 270 . . . . . . 7  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  -> 
( X  -  X
)  =  0 )
85, 7eqtrd 2115 . . . . . 6  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  ->  Y  =  0 )
98ex 113 . . . . 5  |-  ( X  e.  CC  ->  ( Y  =  ( X  -  X )  ->  Y  =  0 ) )
104, 9syl5bi 150 . . . 4  |-  ( X  e.  CC  ->  (
( X  -  X
)  =  Y  ->  Y  =  0 ) )
1110adantr 270 . . 3  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  -  X )  =  Y  ->  Y  =  0 ) )
123, 11sylbird 168 . 2  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  +  Y )  =  X  ->  Y  =  0 ) )
13 oveq2 5571 . . . . 5  |-  ( Y  =  0  ->  ( X  +  Y )  =  ( X  + 
0 ) )
14 addid1 7365 . . . . 5  |-  ( X  e.  CC  ->  ( X  +  0 )  =  X )
1513, 14sylan9eqr 2137 . . . 4  |-  ( ( X  e.  CC  /\  Y  =  0 )  ->  ( X  +  Y )  =  X )
1615ex 113 . . 3  |-  ( X  e.  CC  ->  ( Y  =  0  ->  ( X  +  Y )  =  X ) )
1716adantr 270 . 2  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( Y  =  0  ->  ( X  +  Y )  =  X ) )
1812, 17impbid 127 1  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  +  Y )  =  X  <-> 
Y  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434  (class class class)co 5563   CCcc 7093   0cc0 7095    + caddc 7098    - cmin 7398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308  ax-resscn 7182  ax-1cn 7183  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-sub 7400
This theorem is referenced by:  addn0nid  7597
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