ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnegexlem1 Unicode version

Theorem cnegexlem1 7655
Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7658. (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 7452 . . . 4  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
213ad2ant1 964 . . 3  |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  E. x  e.  RR  ( A  +  x )  =  0 )
3 recn 7473 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 7473 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
5 oveq2 5660 . . . . . . . . . . 11  |-  ( ( A  +  B )  =  ( A  +  C )  ->  (
x  +  ( A  +  B ) )  =  ( x  +  ( A  +  C
) ) )
6 simpr 108 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  x  e.  CC )
7 simpll 496 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  A  e.  CC )
8 simplrl 502 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  B  e.  CC )
96, 7, 8addassd 7508 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  +  B )  =  ( x  +  ( A  +  B
) ) )
10 simplrr 503 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  C  e.  CC )
116, 7, 10addassd 7508 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( x  +  A
)  +  C )  =  ( x  +  ( A  +  C
) ) )
129, 11eqeq12d 2102 . . . . . . . . . . 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 154 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  B
)  =  ( A  +  C )  -> 
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C ) ) )
1413adantr 270 . . . . . . . . 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 7617 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  +  x
)  =  ( x  +  A ) )
1615eqeq1d 2096 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( A  +  x )  =  0  <-> 
( x  +  A
)  =  0 ) )
1716adantlr 461 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  <->  (
x  +  A )  =  0 ) )
18 oveq1 5659 . . . . . . . . . . . . . . 15  |-  ( ( x  +  A )  =  0  ->  (
( x  +  A
)  +  B )  =  ( 0  +  B ) )
19 oveq1 5659 . . . . . . . . . . . . . . 15  |-  ( ( x  +  A )  =  0  ->  (
( x  +  A
)  +  C )  =  ( 0  +  C ) )
2018, 19eqeq12d 2102 . . . . . . . . . . . . . 14  |-  ( ( x  +  A )  =  0  ->  (
( ( x  +  A )  +  B
)  =  ( ( x  +  A )  +  C )  <->  ( 0  +  B )  =  ( 0  +  C
) ) )
2120adantl 271 . . . . . . . . . . . . 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 7619 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CC  ->  (
0  +  B )  =  B )
23 addid2 7619 . . . . . . . . . . . . . . . 16  |-  ( C  e.  CC  ->  (
0  +  C )  =  C )
2422, 23eqeqan12d 2103 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( 0  +  B )  =  ( 0  +  C )  <-> 
B  =  C ) )
2524adantl 271 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( (
0  +  B )  =  ( 0  +  C )  <->  B  =  C ) )
2625ad2antrr 472 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( x  +  A
)  =  0 )  ->  ( ( 0  +  B )  =  ( 0  +  C
)  <->  B  =  C
) )
2721, 26bitrd 186 . . . . . . . . . . . 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 113 . . . . . . . . . . 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 148 . . . . . . . . . 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 122 . . . . . . . . 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 147 . . . . . . . 8  |-  ( ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  /\  ( A  +  x
)  =  0 )  ->  ( ( A  +  B )  =  ( A  +  C
)  ->  B  =  C ) )
3231ex 113 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  CC )  ->  (
( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
334, 32sylan2 280 . . . . . 6  |-  ( ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  /\  x  e.  RR )  ->  (
( A  +  x
)  =  0  -> 
( ( A  +  B )  =  ( A  +  C )  ->  B  =  C ) ) )
3433rexlimdva 2489 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( E. x  e.  RR  ( A  +  x )  =  0  ->  (
( A  +  B
)  =  ( A  +  C )  ->  B  =  C )
) )
35343impb 1139 . . . 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 1207 . . 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 5660 . 2  |-  ( B  =  C  ->  ( A  +  B )  =  ( A  +  C ) )
3937, 38impbid1 140 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360  (class class class)co 5652   CCcc 7346   RRcr 7347   0cc0 7348    + caddc 7351
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-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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7435  ax-1cn 7436  ax-icn 7438  ax-addcl 7439  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  cnegexlem3  7657
  Copyright terms: Public domain W3C validator