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Theorem cnegexlem2 8146
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 8148. (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 7962 . 2  |-  0  e.  CC
2 cnre 7966 . 2  |-  ( 0  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) ) )
3 ax-rnegex 7933 . . . . . 6  |-  ( x  e.  RR  ->  E. z  e.  RR  ( x  +  z )  =  0 )
43adantr 276 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  E. z  e.  RR  ( x  +  z
)  =  0 )
5 recn 7957 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
6 ax-icn 7919 . . . . . . . . . . . 12  |-  _i  e.  CC
7 recn 7957 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  e.  CC )
8 mulcl 7951 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
96, 7, 8sylancr 414 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
10 recn 7957 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
11 addlid 8109 . . . . . . . . . . . . . . 15  |-  ( z  e.  CC  ->  (
0  +  z )  =  z )
12113ad2ant3 1021 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( 0  +  z )  =  z )
1312adantr 276 . . . . . . . . . . . . 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 5895 . . . . . . . . . . . . . . 15  |-  ( ( x  +  z )  =  0  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( 0  +  ( _i  x.  y
) ) )
1514ad2antrl 490 . . . . . . . . . . . . . 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 8129 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  z  e.  CC  /\  (
_i  x.  y )  e.  CC )  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
17163com23 1210 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( ( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
18 oveq1 5895 . . . . . . . . . . . . . . . . 17  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  +  z )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
1918eqcomd 2193 . . . . . . . . . . . . . . . 16  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
( x  +  ( _i  x.  y ) )  +  z )  =  ( 0  +  z ) )
2017, 19sylan9eq 2240 . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . 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 addlid 8109 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  y )  e.  CC  ->  (
0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
23223ad2ant2 1020 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( 0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
2423adantr 276 . . . . . . . . . . . . . 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 2228 . . . . . . . . . . . . 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 2222 . . . . . . . . . . . 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 115 . . . . . . . . . . 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 1290 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  z  e.  RR )  ->  (
( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) )  ->  z  =  ( _i  x.  y
) ) )
29283expa 1204 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  ->  ( ( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y
) ) )  -> 
z  =  ( _i  x.  y ) ) )
3029imp 124 . . . . . . . 8  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  =  ( _i  x.  y
) )
31 simplr 528 . . . . . . . 8  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  e.  RR )
3230, 31eqeltrrd 2265 . . . . . . 7  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  ( _i  x.  y )  e.  RR )
3332exp32 365 . . . . . 6  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  ->  ( ( x  +  z )  =  0  ->  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
_i  x.  y )  e.  RR ) ) )
3433rexlimdva 2604 . . . . 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 2589 . . 3  |-  ( x  e.  RR  ->  ( E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) )  ->  E. y  e.  RR  ( _i  x.  y
)  e.  RR ) )
3736rexlimiv 2598 . 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 104    /\ w3a 979    = wceq 1363    e. wcel 2158   E.wrex 2466  (class class class)co 5888   CCcc 7822   RRcr 7823   0cc0 7824   _ici 7826    + caddc 7827    x. cmul 7829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-resscn 7916  ax-1cn 7917  ax-icn 7919  ax-addcl 7920  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891
This theorem is referenced by:  cnegex  8148
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