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Theorem cnegexlem2 7719
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 7721. (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 7541 . 2  |-  0  e.  CC
2 cnre 7545 . 2  |-  ( 0  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) ) )
3 ax-rnegex 7515 . . . . . 6  |-  ( x  e.  RR  ->  E. z  e.  RR  ( x  +  z )  =  0 )
43adantr 271 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  E. z  e.  RR  ( x  +  z
)  =  0 )
5 recn 7536 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
6 ax-icn 7501 . . . . . . . . . . . 12  |-  _i  e.  CC
7 recn 7536 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  e.  CC )
8 mulcl 7530 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
96, 7, 8sylancr 406 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
10 recn 7536 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
11 addid2 7682 . . . . . . . . . . . . . . 15  |-  ( z  e.  CC  ->  (
0  +  z )  =  z )
12113ad2ant3 967 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( 0  +  z )  =  z )
1312adantr 271 . . . . . . . . . . . . 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 5673 . . . . . . . . . . . . . . 15  |-  ( ( x  +  z )  =  0  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( 0  +  ( _i  x.  y
) ) )
1514ad2antrl 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  +  ( _i  x.  y
) ) )
16 add32 7702 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  CC  /\  z  e.  CC  /\  (
_i  x.  y )  e.  CC )  ->  (
( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
17163com23 1150 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( ( x  +  z )  +  ( _i  x.  y ) )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
18 oveq1 5673 . . . . . . . . . . . . . . . . 17  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  +  z )  =  ( ( x  +  ( _i  x.  y ) )  +  z ) )
1918eqcomd 2094 . . . . . . . . . . . . . . . 16  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
( x  +  ( _i  x.  y ) )  +  z )  =  ( 0  +  z ) )
2017, 19sylan9eq 2141 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 7682 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  y )  e.  CC  ->  (
0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
23223ad2ant2 966 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  z  e.  CC )  ->  ( 0  +  ( _i  x.  y ) )  =  ( _i  x.  y ) )
2423adantr 271 . . . . . . . . . . . . . 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 2129 . . . . . . . . . . . . 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 2123 . . . . . . . . . . . 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 1217 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  z  e.  RR )  ->  (
( ( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) )  ->  z  =  ( _i  x.  y
) ) )
29283expa 1144 . . . . . . . . 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 498 . . . . . . . 8  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  z  e.  RR )
3230, 31eqeltrrd 2166 . . . . . . 7  |-  ( ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  /\  (
( x  +  z )  =  0  /\  0  =  ( x  +  ( _i  x.  y ) ) ) )  ->  ( _i  x.  y )  e.  RR )
3332exp32 358 . . . . . 6  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  z  e.  RR )  ->  ( ( x  +  z )  =  0  ->  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
_i  x.  y )  e.  RR ) ) )
3433rexlimdva 2490 . . . . 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 2476 . . 3  |-  ( x  e.  RR  ->  ( E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) )  ->  E. y  e.  RR  ( _i  x.  y
)  e.  RR ) )
3736rexlimiv 2484 . 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 925    = wceq 1290    e. wcel 1439   E.wrex 2361  (class class class)co 5666   CCcc 7409   RRcr 7410   0cc0 7411   _ici 7413    + caddc 7414    x. cmul 7416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-resscn 7498  ax-1cn 7499  ax-icn 7501  ax-addcl 7502  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-i2m1 7511  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669
This theorem is referenced by:  cnegex  7721
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