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Theorem cnegexlem2 8095
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 8097. (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 7912 . 2  |-  0  e.  CC
2 cnre 7916 . 2  |-  ( 0  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) ) )
3 ax-rnegex 7883 . . . . . 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 7907 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
6 ax-icn 7869 . . . . . . . . . . . 12  |-  _i  e.  CC
7 recn 7907 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  e.  CC )
8 mulcl 7901 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
96, 7, 8sylancr 412 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
10 recn 7907 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
11 addid2 8058 . . . . . . . . . . . . . . 15  |-  ( z  e.  CC  ->  (
0  +  z )  =  z )
12113ad2ant3 1015 . . . . . . . . . . . . . 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 5860 . . . . . . . . . . . . . . 15  |-  ( ( x  +  z )  =  0  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( 0  +  ( _i  x.  y
) ) )
1514ad2antrl 487 . . . . . . . . . . . . . 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 8078 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  z  e.  CC  /\  (
_i  x.  y )  e.  CC )  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
17163com23 1204 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( ( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
18 oveq1 5860 . . . . . . . . . . . . . . . . 17  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  +  z )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
1918eqcomd 2176 . . . . . . . . . . . . . . . 16  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
( x  +  ( _i  x.  y ) )  +  z )  =  ( 0  +  z ) )
2017, 19sylan9eq 2223 . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . 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 8058 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  y )  e.  CC  ->  (
0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
23223ad2ant2 1014 . . . . . . . . . . . . . . 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 2211 . . . . . . . . . . . . 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 2205 . . . . . . . . . . . 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 1275 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  z  e.  RR )  ->  (
( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) )  ->  z  =  ( _i  x.  y
) ) )
29283expa 1198 . . . . . . . . 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 525 . . . . . . . 8  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  e.  RR )
3230, 31eqeltrrd 2248 . . . . . . 7  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  ( _i  x.  y )  e.  RR )
3332exp32 363 . . . . . 6  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  ->  ( ( x  +  z )  =  0  ->  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
_i  x.  y )  e.  RR ) ) )
3433rexlimdva 2587 . . . . 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 2572 . . 3  |-  ( x  e.  RR  ->  ( E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) )  ->  E. y  e.  RR  ( _i  x.  y
)  e.  RR ) )
3736rexlimiv 2581 . 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 973    = wceq 1348    e. wcel 2141   E.wrex 2449  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   _ici 7776    + caddc 7777    x. cmul 7779
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856
This theorem is referenced by:  cnegex  8097
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