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Theorem cnegexlem2 6984
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 6986. (Contributed by Eric Schmidt, 22-May-2007.)
Assertion
Ref Expression
cnegexlem2  RR  _i  x.  RR

Proof of Theorem cnegexlem2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0cn 6817 . 2  0  CC
2 cnre 6821 . 2  0  CC  RR  RR  0  +  _i  x.
3 ax-rnegex 6792 . . . . . 6  RR  RR  +  0
43adantr 261 . . . . 5  RR  RR  RR  +  0
5 recn 6812 . . . . . . . . . . 11  RR  CC
6 ax-icn 6778 . . . . . . . . . . . 12  _i  CC
7 recn 6812 . . . . . . . . . . . 12  RR  CC
8 mulcl 6806 . . . . . . . . . . . 12  _i  CC  CC  _i  x.  CC
96, 7, 8sylancr 393 . . . . . . . . . . 11  RR  _i  x.  CC
10 recn 6812 . . . . . . . . . . 11  RR  CC
11 addid2 6949 . . . . . . . . . . . . . . 15  CC 
0  +
12113ad2ant3 926 . . . . . . . . . . . . . 14  CC  _i  x.  CC  CC 
0  +
1312adantr 261 . . . . . . . . . . . . 13  CC  _i  x.  CC  CC  +  0  0  +  _i  x.  0  +
14 oveq1 5462 . . . . . . . . . . . . . . 15  +  0  +  +  _i  x.  0  +  _i  x.
1514ad2antrl 459 . . . . . . . . . . . . . 14  CC  _i  x.  CC  CC  +  0  0  +  _i  x.  +  +  _i  x.  0  +  _i  x.
16 add32 6967 . . . . . . . . . . . . . . . . 17  CC  CC  _i  x.  CC  +  +  _i  x.  +  _i  x.  +
17163com23 1109 . . . . . . . . . . . . . . . 16  CC  _i  x.  CC  CC  +  +  _i  x.  +  _i  x.  +
18 oveq1 5462 . . . . . . . . . . . . . . . . 17  0  +  _i  x. 
0  +  +  _i  x.  +
1918eqcomd 2042 . . . . . . . . . . . . . . . 16  0  +  _i  x.  +  _i  x.  +  0  +
2017, 19sylan9eq 2089 . . . . . . . . . . . . . . 15  CC  _i  x.  CC  CC  0  +  _i  x.  +  +  _i  x.  0  +
2120adantrl 447 . . . . . . . . . . . . . 14  CC  _i  x.  CC  CC  +  0  0  +  _i  x.  +  +  _i  x.  0  +
22 addid2 6949 . . . . . . . . . . . . . . . 16  _i  x.  CC 
0  +  _i  x.  _i  x.
23223ad2ant2 925 . . . . . . . . . . . . . . 15  CC  _i  x.  CC  CC 
0  +  _i  x.  _i  x.
2423adantr 261 . . . . . . . . . . . . . 14  CC  _i  x.  CC  CC  +  0  0  +  _i  x.  0  +  _i  x.  _i  x.
2515, 21, 243eqtr3d 2077 . . . . . . . . . . . . 13  CC  _i  x.  CC  CC  +  0  0  +  _i  x.  0  +  _i  x.
2613, 25eqtr3d 2071 . . . . . . . . . . . 12  CC  _i  x.  CC  CC  +  0  0  +  _i  x.  _i  x.
2726ex 108 . . . . . . . . . . 11  CC  _i  x.  CC  CC  +  0  0  +  _i  x.  _i  x.
285, 9, 10, 27syl3an 1176 . . . . . . . . . 10  RR  RR  RR  +  0  0  +  _i  x.  _i  x.
29283expa 1103 . . . . . . . . 9  RR  RR  RR  +  0  0  +  _i  x.  _i  x.
3029imp 115 . . . . . . . 8  RR  RR  RR  +  0  0  +  _i  x.  _i  x.
31 simplr 482 . . . . . . . 8  RR  RR  RR  +  0  0  +  _i  x.  RR
3230, 31eqeltrrd 2112 . . . . . . 7  RR  RR  RR  +  0  0  +  _i  x.  _i  x.  RR
3332exp32 347 . . . . . 6  RR  RR  RR  +  0  0  +  _i  x.  _i  x.  RR
3433rexlimdva 2427 . . . . 5  RR  RR  RR  +  0  0  +  _i  x.  _i  x.  RR
354, 34mpd 13 . . . 4  RR  RR  0  +  _i  x.  _i  x.  RR
3635reximdva 2415 . . 3  RR  RR  0  +  _i  x.  RR  _i  x.  RR
3736rexlimiv 2421 . 2  RR  RR  0  +  _i  x.  RR  _i  x.  RR
381, 2, 37mp2b 8 1  RR  _i  x.  RR
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390  wrex 2301  (class class class)co 5455   CCcc 6709   RRcr 6710   0cc0 6711   _ici 6713    + caddc 6714    x. cmul 6716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-resscn 6775  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  cnegex  6986
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