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Theorem mul02 8870
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
StepHypRef Expression
1 cnre 8716 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 recn 8707 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
3 ax-icn 8676 . . . . . . . 8  |-  _i  e.  CC
4 recn 8707 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 mulcl 8701 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
63, 4, 5sylancr 647 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
7 0cn 8711 . . . . . . . 8  |-  0  e.  CC
8 adddi 8706 . . . . . . . 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 1269 . . . . . . 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 465 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y
) ) ) )
11 mul02lem2 8869 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
12 mul12 8858 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  _i  e.  CC  /\  y  e.  CC )  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  ( 0  x.  y
) ) )
137, 3, 12mp3an12 1272 . . . . . . . . 9  |-  ( y  e.  CC  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  ( 0  x.  y
) ) )
144, 13syl 17 . . . . . . . 8  |-  ( y  e.  RR  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  ( 0  x.  y
) ) )
15 mul02lem2 8869 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
0  x.  y )  =  0 )
1615oveq2d 5726 . . . . . . . 8  |-  ( y  e.  RR  ->  (
_i  x.  ( 0  x.  y ) )  =  ( _i  x.  0 ) )
1714, 16eqtrd 2285 . . . . . . 7  |-  ( y  e.  RR  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  0 ) )
1811, 17oveqan12d 5729 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
1910, 18eqtrd 2285 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
20 cnre 8716 . . . . . . . 8  |-  ( 0  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) ) )
217, 20ax-mp 10 . . . . . . 7  |-  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) )
22 oveq2 5718 . . . . . . . . . 10  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  0 )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
2322eqeq1d 2261 . . . . . . . . 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 215 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  =  ( x  +  ( _i  x.  y ) )  ->  ( 0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) ) ) )
2524rexlimivv 2634 . . . . . . 7  |-  ( E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) ) )
2621, 25ax-mp 10 . . . . . 6  |-  ( 0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) )
27 0re 8718 . . . . . . 7  |-  0  e.  RR
28 mul02lem2 8869 . . . . . . 7  |-  ( 0  e.  RR  ->  (
0  x.  0 )  =  0 )
2927, 28ax-mp 10 . . . . . 6  |-  ( 0  x.  0 )  =  0
3026, 29eqtr3i 2275 . . . . 5  |-  ( 0  +  ( _i  x.  0 ) )  =  0
3119, 30syl6eq 2301 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  0 )
32 oveq2 5718 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  A )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
3332eqeq1d 2261 . . . 4  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
( 0  x.  A
)  =  0  <->  (
0  x.  ( x  +  ( _i  x.  y ) ) )  =  0 ) )
3431, 33syl5ibrcom 215 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  ->  ( 0  x.  A )  =  0 ) )
3534rexlimivv 2634 . 2  |-  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  A )  =  0 )
361, 35syl 17 1  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2510  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   _ici 8619    + caddc 8620    x. cmul 8622
This theorem is referenced by:  mul01  8871  cnegex2  8874  mul02i  8881  mul02d  8890  bcval5  11208  fsumconst  12129  demoivreALT  12355  cnfldmulg  16238  itg2mulc  18934  dvcmulf  19126  coe0  19469  plymul0or  19493  sineq0  19721  jensen  20115  musumsum  20264  lgsne0  20404  brbtwn2  23707  ax5seglem4  23734  axeuclidlem  23764  axeuclid  23765  axcontlem2  23767  axcontlem4  23769  cnegvex2  24826  expgrowth  26718
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752
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