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Theorem mul02 9792
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 9622 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 recn 9612 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
3 ax-icn 9581 . . . . . . . 8  |-  _i  e.  CC
4 recn 9612 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 mulcl 9606 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
63, 4, 5sylancr 661 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
7 0cn 9618 . . . . . . . 8  |-  0  e.  CC
8 adddi 9611 . . . . . . . 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 1313 . . . . . . 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 475 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y
) ) ) )
11 mul02lem2 9791 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
12 mul12 9780 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  _i  e.  CC  /\  y  e.  CC )  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  ( 0  x.  y
) ) )
137, 3, 12mp3an12 1316 . . . . . . . . 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 9791 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
0  x.  y )  =  0 )
1615oveq2d 6294 . . . . . . . 8  |-  ( y  e.  RR  ->  (
_i  x.  ( 0  x.  y ) )  =  ( _i  x.  0 ) )
1714, 16eqtrd 2443 . . . . . . 7  |-  ( y  e.  RR  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  0 ) )
1811, 17oveqan12d 6297 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
1910, 18eqtrd 2443 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
20 cnre 9622 . . . . . . . 8  |-  ( 0  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) ) )
217, 20ax-mp 5 . . . . . . 7  |-  E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y ) )
22 oveq2 6286 . . . . . . . . . 10  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  0 )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
2322eqeq1d 2404 . . . . . . . . 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 222 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  =  ( x  +  ( _i  x.  y ) )  ->  ( 0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) ) ) )
2524rexlimivv 2901 . . . . . . 7  |-  ( E. x  e.  RR  E. y  e.  RR  0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) ) )
2621, 25ax-mp 5 . . . . . 6  |-  ( 0  x.  0 )  =  ( 0  +  ( _i  x.  0 ) )
27 0re 9626 . . . . . . 7  |-  0  e.  RR
28 mul02lem2 9791 . . . . . . 7  |-  ( 0  e.  RR  ->  (
0  x.  0 )  =  0 )
2927, 28ax-mp 5 . . . . . 6  |-  ( 0  x.  0 )  =  0
3026, 29eqtr3i 2433 . . . . 5  |-  ( 0  +  ( _i  x.  0 ) )  =  0
3119, 30syl6eq 2459 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  0 )
32 oveq2 6286 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  A )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
3332eqeq1d 2404 . . . 4  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
( 0  x.  A
)  =  0  <->  (
0  x.  ( x  +  ( _i  x.  y ) ) )  =  0 ) )
3431, 33syl5ibrcom 222 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  ->  ( 0  x.  A )  =  0 ) )
3534rexlimivv 2901 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522   _ici 9524    + caddc 9525    x. cmul 9527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663
This theorem is referenced by:  mul01  9793  cnegex2  9796  mul02i  9803  mul02d  9812  bcval5  12440  fsumconst  13756  demoivreALT  14145  nnnn0modprm0  14540  cnfldmulg  18770  itg2mulc  22446  dvcmulf  22640  coe0  22945  plymul0or  22969  sineq0  23206  jensen  23644  musumsum  23849  lgsne0  23989  brbtwn2  24625  ax5seglem4  24652  axeuclidlem  24682  axeuclid  24683  axcontlem2  24685  axcontlem4  24687  eulerpartlemb  28813  expgrowth  36088  dvcosax  37091
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