MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mul02 Unicode version

Theorem mul02 8985
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 8829 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 recn 8822 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  CC )
3 ax-icn 8791 . . . . . . . 8  |-  _i  e.  CC
4 recn 8822 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
5 mulcl 8816 . . . . . . . 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 8826 . . . . . . . 8  |-  0  e.  CC
8 adddi 8821 . . . . . . . 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 8984 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
12 mul12 8973 . . . . . . . . . 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 8984 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
0  x.  y )  =  0 )
1615oveq2d 5835 . . . . . . . 8  |-  ( y  e.  RR  ->  (
_i  x.  ( 0  x.  y ) )  =  ( _i  x.  0 ) )
1714, 16eqtrd 2316 . . . . . . 7  |-  ( y  e.  RR  ->  (
0  x.  ( _i  x.  y ) )  =  ( _i  x.  0 ) )
1811, 17oveqan12d 5838 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( 0  x.  x )  +  ( 0  x.  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
1910, 18eqtrd 2316 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  ( 0  +  ( _i  x.  0 ) ) )
20 cnre 8829 . . . . . . . 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 5827 . . . . . . . . . 10  |-  ( 0  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  0 )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
2322eqeq1d 2292 . . . . . . . . 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 2673 . . . . . . 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 8833 . . . . . . 7  |-  0  e.  RR
28 mul02lem2 8984 . . . . . . 7  |-  ( 0  e.  RR  ->  (
0  x.  0 )  =  0 )
2927, 28ax-mp 10 . . . . . 6  |-  ( 0  x.  0 )  =  0
3026, 29eqtr3i 2306 . . . . 5  |-  ( 0  +  ( _i  x.  0 ) )  =  0
3119, 30syl6eq 2332 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( 0  x.  (
x  +  ( _i  x.  y ) ) )  =  0 )
32 oveq2 5827 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
0  x.  A )  =  ( 0  x.  ( x  +  ( _i  x.  y ) ) ) )
3332eqeq1d 2292 . . . 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 2673 . 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 1628    e. wcel 1688   E.wrex 2545  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732   _ici 8734    + caddc 8735    x. cmul 8737
This theorem is referenced by:  mul01  8986  cnegex2  8989  mul02i  8996  mul02d  9005  bcval5  11324  fsumconst  12246  demoivreALT  12475  cnfldmulg  16400  itg2mulc  19096  dvcmulf  19288  coe0  19631  plymul0or  19655  sineq0  19883  jensen  20277  musumsum  20426  lgsne0  20566  brbtwn2  23940  ax5seglem4  23967  axeuclidlem  23997  axeuclid  23998  axcontlem2  24000  axcontlem4  24002  cnegvex2  25059  expgrowth  26951
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-ltxr 8867
  Copyright terms: Public domain W3C validator