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Theorem cnegexlem1 7249
Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7252. (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 7051 . . . 4  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
213ad2ant1 936 . . 3  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  RR  ( A  +  x )  =  0 )
3 recn 7072 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 7072 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
5 oveq2 5548 . . . . . . . . . . 11  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
6 simpr 107 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  x  e.  CC )
7 simpll 489 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  A  e.  CC )
8 simplrl 495 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  B  e.  CC )
96, 7, 8addassd 7107 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  +  B )  =  ( x  +  ( A  +  B
) ) )
10 simplrr 496 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  C  e.  CC )
116, 7, 10addassd 7107 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  +  C )  =  ( x  +  ( A  +  C
) ) )
129, 11eqeq12d 2070 . . . . . . . . . . 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 149 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  -> 
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C ) ) )
1413adantr 265 . . . . . . . . 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 7211 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  +  x
)  =  ( x  +  A ) )
1615eqeq1d 2064 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( A  +  x )  =  0  <-> 
( x  +  A
)  =  0 ) )
1716adantlr 454 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  <->  (
x  +  A )  =  0 ) )
18 oveq1 5547 . . . . . . . . . . . . . . 15  |-  ( ( x  +  A )  =  0  ->  (
( x  +  A
)  +  B )  =  ( 0  +  B ) )
19 oveq1 5547 . . . . . . . . . . . . . . 15  |-  ( ( x  +  A )  =  0  ->  (
( x  +  A
)  +  C )  =  ( 0  +  C ) )
2018, 19eqeq12d 2070 . . . . . . . . . . . . . 14  |-  ( ( x  +  A )  =  0  ->  (
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C )  <->  ( 0  +  B )  =  ( 0  +  C
) ) )
2120adantl 266 . . . . . . . . . . . . 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 7213 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CC  ->  (
0  +  B )  =  B )
23 addid2 7213 . . . . . . . . . . . . . . . 16  |-  ( C  e.  CC  ->  (
0  +  C )  =  C )
2422, 23eqeqan12d 2071 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( 0  +  B )  =  ( 0  +  C )  <-> 
B  =  C ) )
2524adantl 266 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( (
0  +  B )  =  ( 0  +  C )  <->  B  =  C ) )
2625ad2antrr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( x  +  A
)  =  0 )  ->  ( ( 0  +  B )  =  ( 0  +  C
)  <->  B  =  C
) )
2721, 26bitrd 181 . . . . . . . . . . . 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 112 . . . . . . . . . . 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 143 . . . . . . . . . 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 119 . . . . . . . . 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 142 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( A  +  x
)  =  0 )  ->  ( ( A  +  B )  =  ( A  +  C
)  ->  B  =  C ) )
3231ex 112 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
334, 32sylan2 274 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  RR )  ->  (
( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
3433rexlimdva 2450 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( E. x  e.  RR  ( A  +  x )  =  0  ->  (
( A  +  B
)  =  ( A  +  C )  ->  B  =  C )
) )
35343impb 1111 . . . 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 1179 . . 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 5548 . 2  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
3937, 38impbid1 134 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 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947    + caddc 6950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by:  cnegexlem3  7251
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