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Theorem cnegexlem1 7937
Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7940. (Contributed by Eric Schmidt, 22-May-2007.)
Assertion
Ref Expression
cnegexlem1  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  <->  B  =  C ) )

Proof of Theorem cnegexlem1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ax-rnegex 7729 . . . 4  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
213ad2ant1 1002 . . 3  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  RR  ( A  +  x )  =  0 )
3 recn 7753 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 7753 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
5 oveq2 5782 . . . . . . . . . . 11  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
6 simpr 109 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  x  e.  CC )
7 simpll 518 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  A  e.  CC )
8 simplrl 524 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  B  e.  CC )
96, 7, 8addassd 7788 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  +  B )  =  ( x  +  ( A  +  B
) ) )
10 simplrr 525 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  C  e.  CC )
116, 7, 10addassd 7788 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  +  C )  =  ( x  +  ( A  +  C
) ) )
129, 11eqeq12d 2154 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C )  <->  ( x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C ) ) ) )
135, 12syl5ibr 155 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  -> 
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C ) ) )
1413adantr 274 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( A  +  x
)  =  0 )  ->  ( ( A  +  B )  =  ( A  +  C
)  ->  ( (
x  +  A )  +  B )  =  ( ( x  +  A )  +  C
) ) )
15 addcom 7899 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  +  x
)  =  ( x  +  A ) )
1615eqeq1d 2148 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( A  +  x )  =  0  <-> 
( x  +  A
)  =  0 ) )
1716adantlr 468 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  <->  (
x  +  A )  =  0 ) )
18 oveq1 5781 . . . . . . . . . . . . . . 15  |-  ( ( x  +  A )  =  0  ->  (
( x  +  A
)  +  B )  =  ( 0  +  B ) )
19 oveq1 5781 . . . . . . . . . . . . . . 15  |-  ( ( x  +  A )  =  0  ->  (
( x  +  A
)  +  C )  =  ( 0  +  C ) )
2018, 19eqeq12d 2154 . . . . . . . . . . . . . 14  |-  ( ( x  +  A )  =  0  ->  (
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C )  <->  ( 0  +  B )  =  ( 0  +  C
) ) )
2120adantl 275 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( x  +  A
)  =  0 )  ->  ( ( ( x  +  A )  +  B )  =  ( ( x  +  A )  +  C
)  <->  ( 0  +  B )  =  ( 0  +  C ) ) )
22 addid2 7901 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CC  ->  (
0  +  B )  =  B )
23 addid2 7901 . . . . . . . . . . . . . . . 16  |-  ( C  e.  CC  ->  (
0  +  C )  =  C )
2422, 23eqeqan12d 2155 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( 0  +  B )  =  ( 0  +  C )  <-> 
B  =  C ) )
2524adantl 275 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( (
0  +  B )  =  ( 0  +  C )  <->  B  =  C ) )
2625ad2antrr 479 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( x  +  A
)  =  0 )  ->  ( ( 0  +  B )  =  ( 0  +  C
)  <->  B  =  C
) )
2721, 26bitrd 187 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( x  +  A
)  =  0 )  ->  ( ( ( x  +  A )  +  B )  =  ( ( x  +  A )  +  C
)  <->  B  =  C
) )
2827ex 114 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  =  0  -> 
( ( ( x  +  A )  +  B )  =  ( ( x  +  A
)  +  C )  <-> 
B  =  C ) ) )
2917, 28sylbid 149 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  -> 
( ( ( x  +  A )  +  B )  =  ( ( x  +  A
)  +  C )  <-> 
B  =  C ) ) )
3029imp 123 . . . . . . . . 9  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( A  +  x
)  =  0 )  ->  ( ( ( x  +  A )  +  B )  =  ( ( x  +  A )  +  C
)  <->  B  =  C
) )
3114, 30sylibd 148 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( A  +  x
)  =  0 )  ->  ( ( A  +  B )  =  ( A  +  C
)  ->  B  =  C ) )
3231ex 114 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
334, 32sylan2 284 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  RR )  ->  (
( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
3433rexlimdva 2549 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( E. x  e.  RR  ( A  +  x )  =  0  ->  (
( A  +  B
)  =  ( A  +  C )  ->  B  =  C )
) )
35343impb 1177 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( E. x  e.  RR  ( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
363, 35syl3an1 1249 . . 3  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  ( E. x  e.  RR  ( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
372, 36mpd 13 . 2  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  ->  B  =  C )
)
38 oveq2 5782 . 2  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
3937, 38impbid1 141 1  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620    + caddc 7623
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-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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  cnegexlem3  7939
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