ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnegexlem2 Unicode version

Theorem cnegexlem2 7945
Description: Existence of a real number which produces a real number when multiplied by  _i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 7947. (Contributed by Eric Schmidt, 22-May-2007.)
Assertion
Ref Expression
cnegexlem2  |-  E. y  e.  RR  ( _i  x.  y )  e.  RR

Proof of Theorem cnegexlem2
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0cn 7765 . 2  |-  0  e.  CC
2 cnre 7769 . 2  |-  ( 0  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) ) )
3 ax-rnegex 7736 . . . . . 6  |-  ( x  e.  RR  ->  E. z  e.  RR  ( x  +  z )  =  0 )
43adantr 274 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  E. z  e.  RR  ( x  +  z
)  =  0 )
5 recn 7760 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
6 ax-icn 7722 . . . . . . . . . . . 12  |-  _i  e.  CC
7 recn 7760 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  e.  CC )
8 mulcl 7754 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
96, 7, 8sylancr 410 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
10 recn 7760 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
11 addid2 7908 . . . . . . . . . . . . . . 15  |-  ( z  e.  CC  ->  (
0  +  z )  =  z )
12113ad2ant3 1004 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( 0  +  z )  =  z )
1312adantr 274 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  /\  ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  (
0  +  z )  =  z )
14 oveq1 5781 . . . . . . . . . . . . . . 15  |-  ( ( x  +  z )  =  0  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( 0  +  ( _i  x.  y
) ) )
1514ad2antrl 481 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  /\  ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( 0  +  ( _i  x.  y
) ) )
16 add32 7928 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  z  e.  CC  /\  (
_i  x.  y )  e.  CC )  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
17163com23 1187 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( ( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
18 oveq1 5781 . . . . . . . . . . . . . . . . 17  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  +  z )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
1918eqcomd 2145 . . . . . . . . . . . . . . . 16  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
( x  +  ( _i  x.  y ) )  +  z )  =  ( 0  +  z ) )
2017, 19sylan9eq 2192 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  /\  0  =  (
x  +  ( _i  x.  y ) ) )  ->  ( (
x  +  z )  +  ( _i  x.  y ) )  =  ( 0  +  z ) )
2120adantrl 469 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  /\  ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( 0  +  z ) )
22 addid2 7908 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  y )  e.  CC  ->  (
0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
23223ad2ant2 1003 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( 0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
2423adantr 274 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  /\  ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  (
0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
2515, 21, 243eqtr3d 2180 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  /\  ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  (
0  +  z )  =  ( _i  x.  y ) )
2613, 25eqtr3d 2174 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  /\  ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  =  ( _i  x.  y ) )
2726ex 114 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) )  ->  z  =  ( _i  x.  y ) ) )
285, 9, 10, 27syl3an 1258 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  z  e.  RR )  ->  (
( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) )  ->  z  =  ( _i  x.  y
) ) )
29283expa 1181 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  ->  ( ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y
) ) )  -> 
z  =  ( _i  x.  y ) ) )
3029imp 123 . . . . . . . 8  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  =  ( _i  x.  y
) )
31 simplr 519 . . . . . . . 8  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  e.  RR )
3230, 31eqeltrrd 2217 . . . . . . 7  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  ( _i  x.  y )  e.  RR )
3332exp32 362 . . . . . 6  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  ->  ( ( x  +  z )  =  0  ->  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
_i  x.  y )  e.  RR ) ) )
3433rexlimdva 2549 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( E. z  e.  RR  ( x  +  z )  =  0  ->  ( 0  =  ( x  +  ( _i  x.  y ) )  ->  ( _i  x.  y )  e.  RR ) ) )
354, 34mpd 13 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  =  ( x  +  ( _i  x.  y ) )  ->  ( _i  x.  y )  e.  RR ) )
3635reximdva 2534 . . 3  |-  ( x  e.  RR  ->  ( E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) )  ->  E. y  e.  RR  ( _i  x.  y
)  e.  RR ) )
3736rexlimiv 2543 . 2  |-  ( E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y
) )  ->  E. y  e.  RR  ( _i  x.  y )  e.  RR )
381, 2, 37mp2b 8 1  |-  E. y  e.  RR  ( _i  x.  y )  e.  RR
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627   _ici 7629    + caddc 7630    x. cmul 7632
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 7719  ax-1cn 7720  ax-icn 7722  ax-addcl 7723  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-i2m1 7732  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738
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:  cnegex  7947
  Copyright terms: Public domain W3C validator