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Theorem mul02 9233
Description: Multiplication by  0. Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
mul02  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )

Proof of Theorem mul02
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 9076 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 recn 9069 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
3 ax-icn 9038 . . . . . . . 8  |-  _i  e.  CC
4 recn 9069 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 mulcl 9063 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
63, 4, 5sylancr 645 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
7 0cn 9073 . . . . . . . 8  |-  0  e.  CC
8 adddi 9068 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  x  e.  CC  /\  (
_i  x.  y )  e.  CC )  ->  (
0  x.  ( x  +  ( _i  x.  y ) ) )  =  ( ( 0  x.  x )  +  ( 0  x.  (
_i  x.  y )
) ) )
97, 8mp3an1 1266 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC )  ->  ( 0  x.  ( x  +  ( _i  x.  y ) ) )  =  ( ( 0  x.  x
)  +  ( 0  x.  ( _i  x.  y ) ) ) )
102, 6, 9syl2an 464 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y
) ) ) )
11 mul02lem2 9232 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
12 mul12 9221 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  _i  e.  CC  /\  y  e.  CC )  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  ( 0  x.  y
) ) )
137, 3, 12mp3an12 1269 . . . . . . . . 9  |-  ( y  e.  CC  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  ( 0  x.  y
) ) )
144, 13syl 16 . . . . . . . 8  |-  ( y  e.  RR  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  ( 0  x.  y
) ) )
15 mul02lem2 9232 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
0  x.  y )  =  0 )
1615oveq2d 6088 . . . . . . . 8  |-  ( y  e.  RR  ->  (
_i  x.  ( 0  x.  y ) )  =  ( _i  x.  0 ) )
1714, 16eqtrd 2467 . . . . . . 7  |-  ( y  e.  RR  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  0 ) )
1811, 17oveqan12d 6091 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
1910, 18eqtrd 2467 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
20 cnre 9076 . . . . . . . 8  |-  ( 0  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) ) )
217, 20ax-mp 8 . . . . . . 7  |-  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) )
22 oveq2 6080 . . . . . . . . . 10  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  0 )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
2322eqeq1d 2443 . . . . . . . . 9  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
( 0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) )  <->  ( 0  x.  ( x  +  ( _i  x.  y
) ) )  =  ( 0  +  ( _i  x.  0 ) ) ) )
2419, 23syl5ibrcom 214 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  =  ( x  +  ( _i  x.  y ) )  ->  ( 0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) ) ) )
2524rexlimivv 2827 . . . . . . 7  |-  ( E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) ) )
2621, 25ax-mp 8 . . . . . 6  |-  ( 0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) )
27 0re 9080 . . . . . . 7  |-  0  e.  RR
28 mul02lem2 9232 . . . . . . 7  |-  ( 0  e.  RR  ->  (
0  x.  0 )  =  0 )
2927, 28ax-mp 8 . . . . . 6  |-  ( 0  x.  0 )  =  0
3026, 29eqtr3i 2457 . . . . 5  |-  ( 0  +  ( _i  x.  0 ) )  =  0
3119, 30syl6eq 2483 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  0 )
32 oveq2 6080 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  A )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
3332eqeq1d 2443 . . . 4  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
( 0  x.  A
)  =  0  <->  (
0  x.  ( x  +  ( _i  x.  y ) ) )  =  0 ) )
3431, 33syl5ibrcom 214 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  ->  ( 0  x.  A )  =  0 ) )
3534rexlimivv 2827 . 2  |-  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  A )  =  0 )
361, 35syl 16 1  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979   _ici 8981    + caddc 8982    x. cmul 8984
This theorem is referenced by:  mul01  9234  cnegex2  9237  mul02i  9244  mul02d  9253  bcval5  11597  fsumconst  12561  demoivreALT  12790  cnfldmulg  16721  itg2mulc  19627  dvcmulf  19819  coe0  20162  plymul0or  20186  sineq0  20417  jensen  20815  musumsum  20965  lgsne0  21105  brbtwn2  25792  ax5seglem4  25819  axeuclidlem  25849  axeuclid  25850  axcontlem2  25852  axcontlem4  25854  expgrowth  27467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-ltxr 9114
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