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Theorem addid0 8265
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 8221 . . 3  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  -  X )  =  Y  <-> 
( X  +  Y
)  =  X ) )
4 eqcom 2166 . . . . 5  |-  ( ( X  -  X )  =  Y  <->  Y  =  ( X  -  X
) )
5 simpr 109 . . . . . . 7  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  ->  Y  =  ( X  -  X ) )
6 subid 8111 . . . . . . . 8  |-  ( X  e.  CC  ->  ( X  -  X )  =  0 )
76adantr 274 . . . . . . 7  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  -> 
( X  -  X
)  =  0 )
85, 7eqtrd 2197 . . . . . 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 5847 . . . . 5  |-  ( Y  =  0  ->  ( X  +  Y )  =  ( X  + 
0 ) )
14 addid1 8030 . . . . 5  |-  ( X  e.  CC  ->  ( X  +  0 )  =  X )
1513, 14sylan9eqr 2219 . . . 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 1342    e. wcel 2135  (class class class)co 5839   CCcc 7745   0cc0 7747    + caddc 7750    - cmin 8063
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-setind 4511  ax-resscn 7839  ax-1cn 7840  ax-icn 7842  ax-addcl 7843  ax-addrcl 7844  ax-mulcl 7845  ax-addcom 7847  ax-addass 7849  ax-distr 7851  ax-i2m1 7852  ax-0id 7855  ax-rnegex 7856  ax-cnre 7858
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-iota 5150  df-fun 5187  df-fv 5193  df-riota 5795  df-ov 5842  df-oprab 5843  df-mpo 5844  df-sub 8065
This theorem is referenced by:  addn0nid  8266
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