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Theorem cnegexlem2 7648
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 7650. (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 7470 . 2  |-  0  e.  CC
2 cnre 7474 . 2  |-  ( 0  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) ) )
3 ax-rnegex 7444 . . . . . 6  |-  ( x  e.  RR  ->  E. z  e.  RR  ( x  +  z )  =  0 )
43adantr 270 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  E. z  e.  RR  ( x  +  z
)  =  0 )
5 recn 7465 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
6 ax-icn 7430 . . . . . . . . . . . 12  |-  _i  e.  CC
7 recn 7465 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  e.  CC )
8 mulcl 7459 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
96, 7, 8sylancr 405 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
10 recn 7465 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
11 addid2 7611 . . . . . . . . . . . . . . 15  |-  ( z  e.  CC  ->  (
0  +  z )  =  z )
12113ad2ant3 966 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( 0  +  z )  =  z )
1312adantr 270 . . . . . . . . . . . . 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 5651 . . . . . . . . . . . . . . 15  |-  ( ( x  +  z )  =  0  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( 0  +  ( _i  x.  y
) ) )
1514ad2antrl 474 . . . . . . . . . . . . . 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 7631 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  z  e.  CC  /\  (
_i  x.  y )  e.  CC )  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
17163com23 1149 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( ( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
18 oveq1 5651 . . . . . . . . . . . . . . . . 17  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  +  z )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
1918eqcomd 2093 . . . . . . . . . . . . . . . 16  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
( x  +  ( _i  x.  y ) )  +  z )  =  ( 0  +  z ) )
2017, 19sylan9eq 2140 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 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 7611 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  y )  e.  CC  ->  (
0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
23223ad2ant2 965 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( 0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
2423adantr 270 . . . . . . . . . . . . . 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 2128 . . . . . . . . . . . . 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 2122 . . . . . . . . . . . 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 113 . . . . . . . . . . 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 1216 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  z  e.  RR )  ->  (
( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) )  ->  z  =  ( _i  x.  y
) ) )
29283expa 1143 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  ->  ( ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y
) ) )  -> 
z  =  ( _i  x.  y ) ) )
3029imp 122 . . . . . . . 8  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  =  ( _i  x.  y
) )
31 simplr 497 . . . . . . . 8  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  e.  RR )
3230, 31eqeltrrd 2165 . . . . . . 7  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  ( _i  x.  y )  e.  RR )
3332exp32 357 . . . . . 6  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  ->  ( ( x  +  z )  =  0  ->  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
_i  x.  y )  e.  RR ) ) )
3433rexlimdva 2489 . . . . 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 2475 . . 3  |-  ( x  e.  RR  ->  ( E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) )  ->  E. y  e.  RR  ( _i  x.  y
)  e.  RR ) )
3736rexlimiv 2483 . 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 102    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360  (class class class)co 5644   CCcc 7338   RRcr 7339   0cc0 7340   _ici 7342    + caddc 7343    x. cmul 7345
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 7427  ax-1cn 7428  ax-icn 7430  ax-addcl 7431  ax-mulcl 7433  ax-addcom 7435  ax-addass 7437  ax-i2m1 7440  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446
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 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-iota 4975  df-fv 5018  df-ov 5647
This theorem is referenced by:  cnegex  7650
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