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Theorem addid0 7830
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 7790 . . 3  |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  -  X )  =  Y  <-> 
( X  +  Y
)  =  X ) )
4 eqcom 2090 . . . . 5  |-  ( ( X  -  X )  =  Y  <->  Y  =  ( X  -  X
) )
5 simpr 108 . . . . . . 7  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  ->  Y  =  ( X  -  X ) )
6 subid 7680 . . . . . . . 8  |-  ( X  e.  CC  ->  ( X  -  X )  =  0 )
76adantr 270 . . . . . . 7  |-  ( ( X  e.  CC  /\  Y  =  ( X  -  X ) )  -> 
( X  -  X
)  =  0 )
85, 7eqtrd 2120 . . . . . 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 5642 . . . . 5  |-  ( Y  =  0  ->  ( X  +  Y )  =  ( X  + 
0 ) )
14 addid1 7599 . . . . 5  |-  ( X  e.  CC  ->  ( X  +  0 )  =  X )
1513, 14sylan9eqr 2142 . . . 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 1289    e. wcel 1438  (class class class)co 5634   CCcc 7327   0cc0 7329    + caddc 7332    - cmin 7632
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sub 7634
This theorem is referenced by:  addn0nid  7831
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