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Theorem mul02 9746
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 9581 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 recn 9571 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
3 ax-icn 9540 . . . . . . . 8  |-  _i  e.  CC
4 recn 9571 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 mulcl 9565 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
63, 4, 5sylancr 663 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
7 0cn 9577 . . . . . . . 8  |-  0  e.  CC
8 adddi 9570 . . . . . . . 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 1306 . . . . . . 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 477 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y
) ) ) )
11 mul02lem2 9745 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
12 mul12 9734 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  _i  e.  CC  /\  y  e.  CC )  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  ( 0  x.  y
) ) )
137, 3, 12mp3an12 1309 . . . . . . . . 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 9745 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
0  x.  y )  =  0 )
1615oveq2d 6291 . . . . . . . 8  |-  ( y  e.  RR  ->  (
_i  x.  ( 0  x.  y ) )  =  ( _i  x.  0 ) )
1714, 16eqtrd 2501 . . . . . . 7  |-  ( y  e.  RR  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  0 ) )
1811, 17oveqan12d 6294 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
1910, 18eqtrd 2501 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
20 cnre 9581 . . . . . . . 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 6283 . . . . . . . . . 10  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  0 )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
2322eqeq1d 2462 . . . . . . . . 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 2953 . . . . . . 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 9585 . . . . . . 7  |-  0  e.  RR
28 mul02lem2 9745 . . . . . . 7  |-  ( 0  e.  RR  ->  (
0  x.  0 )  =  0 )
2927, 28ax-mp 5 . . . . . 6  |-  ( 0  x.  0 )  =  0
3026, 29eqtr3i 2491 . . . . 5  |-  ( 0  +  ( _i  x.  0 ) )  =  0
3119, 30syl6eq 2517 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  0 )
32 oveq2 6283 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  A )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
3332eqeq1d 2462 . . . 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 2953 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2808  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   _ici 9483    + caddc 9484    x. cmul 9486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622
This theorem is referenced by:  mul01  9747  cnegex2  9750  mul02i  9757  mul02d  9766  bcval5  12351  fsumconst  13554  demoivreALT  13786  nnnn0modprm0  14179  cnfldmulg  18214  itg2mulc  21882  dvcmulf  22076  coe0  22380  plymul0or  22404  sineq0  22640  jensen  23039  musumsum  23189  lgsne0  23329  brbtwn2  23877  ax5seglem4  23904  axeuclidlem  23934  axeuclid  23935  axcontlem2  23937  axcontlem4  23939  eulerpartlemb  27933  expgrowth  30795  dvcosax  31211
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