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Theorem cnegexlem1 8145
Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8148. (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 7933 . . . 4  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
213ad2ant1 1019 . . 3  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  RR  ( A  +  x )  =  0 )
3 recn 7957 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 7957 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
5 oveq2 5896 . . . . . . . . . . 11  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
6 simpr 110 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  x  e.  CC )
7 simpll 527 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  A  e.  CC )
8 simplrl 535 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  B  e.  CC )
96, 7, 8addassd 7993 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  +  B )  =  ( x  +  ( A  +  B
) ) )
10 simplrr 536 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  C  e.  CC )
116, 7, 10addassd 7993 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  +  C )  =  ( x  +  ( A  +  C
) ) )
129, 11eqeq12d 2202 . . . . . . . . . . 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, 12imbitrrid 156 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  -> 
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C ) ) )
1413adantr 276 . . . . . . . . 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 8107 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  +  x
)  =  ( x  +  A ) )
1615eqeq1d 2196 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( A  +  x )  =  0  <-> 
( x  +  A
)  =  0 ) )
1716adantlr 477 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  <->  (
x  +  A )  =  0 ) )
18 oveq1 5895 . . . . . . . . . . . . . . 15  |-  ( ( x  +  A )  =  0  ->  (
( x  +  A
)  +  B )  =  ( 0  +  B ) )
19 oveq1 5895 . . . . . . . . . . . . . . 15  |-  ( ( x  +  A )  =  0  ->  (
( x  +  A
)  +  C )  =  ( 0  +  C ) )
2018, 19eqeq12d 2202 . . . . . . . . . . . . . 14  |-  ( ( x  +  A )  =  0  ->  (
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C )  <->  ( 0  +  B )  =  ( 0  +  C
) ) )
2120adantl 277 . . . . . . . . . . . . 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 addlid 8109 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CC  ->  (
0  +  B )  =  B )
23 addlid 8109 . . . . . . . . . . . . . . . 16  |-  ( C  e.  CC  ->  (
0  +  C )  =  C )
2422, 23eqeqan12d 2203 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( 0  +  B )  =  ( 0  +  C )  <-> 
B  =  C ) )
2524adantl 277 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( (
0  +  B )  =  ( 0  +  C )  <->  B  =  C ) )
2625ad2antrr 488 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( x  +  A
)  =  0 )  ->  ( ( 0  +  B )  =  ( 0  +  C
)  <->  B  =  C
) )
2721, 26bitrd 188 . . . . . . . . . . . 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 115 . . . . . . . . . . 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 150 . . . . . . . . . 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 124 . . . . . . . . 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 149 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( A  +  x
)  =  0 )  ->  ( ( A  +  B )  =  ( A  +  C
)  ->  B  =  C ) )
3231ex 115 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
334, 32sylan2 286 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  RR )  ->  (
( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
3433rexlimdva 2604 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( E. x  e.  RR  ( A  +  x )  =  0  ->  (
( A  +  B
)  =  ( A  +  C )  ->  B  =  C )
) )
35343impb 1200 . . . 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 1281 . . 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 5896 . 2  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
3937, 38impbid1 142 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 104    <-> wb 105    /\ w3a 979    = wceq 1363    e. wcel 2158   E.wrex 2466  (class class class)co 5888   CCcc 7822   RRcr 7823   0cc0 7824    + caddc 7827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-resscn 7916  ax-1cn 7917  ax-icn 7919  ax-addcl 7920  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0id 7932  ax-rnegex 7933
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891
This theorem is referenced by:  cnegexlem3  8147
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