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Theorem mulid1 9582
Description:  1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
mulid1  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )

Proof of Theorem mulid1
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 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
3 ax-icn 9540 . . . . . . 7  |-  _i  e.  CC
4 recn 9571 . . . . . . 7  |-  ( y  e.  RR  ->  y  e.  CC )
5 mulcl 9565 . . . . . . 7  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
63, 4, 5sylancr 661 . . . . . 6  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
7 ax-1cn 9539 . . . . . . 7  |-  1  e.  CC
8 adddir 9576 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC  /\  1  e.  CC )  ->  ( ( x  +  ( _i  x.  y
) )  x.  1 )  =  ( ( x  x.  1 )  +  ( ( _i  x.  y )  x.  1 ) ) )
97, 8mp3an3 1311 . . . . . 6  |-  ( ( x  e.  CC  /\  ( _i  x.  y
)  e.  CC )  ->  ( ( x  +  ( _i  x.  y ) )  x.  1 )  =  ( ( x  x.  1 )  +  ( ( _i  x.  y )  x.  1 ) ) )
102, 6, 9syl2an 475 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( _i  x.  y
) )  x.  1 )  =  ( ( x  x.  1 )  +  ( ( _i  x.  y )  x.  1 ) ) )
11 ax-1rid 9551 . . . . . 6  |-  ( x  e.  RR  ->  (
x  x.  1 )  =  x )
12 mulass 9569 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  y  e.  CC  /\  1  e.  CC )  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  ( y  x.  1 ) ) )
133, 7, 12mp3an13 1313 . . . . . . . 8  |-  ( y  e.  CC  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  ( y  x.  1 ) ) )
144, 13syl 16 . . . . . . 7  |-  ( y  e.  RR  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  ( y  x.  1 ) ) )
15 ax-1rid 9551 . . . . . . . 8  |-  ( y  e.  RR  ->  (
y  x.  1 )  =  y )
1615oveq2d 6286 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  ( y  x.  1 ) )  =  ( _i  x.  y
) )
1714, 16eqtrd 2495 . . . . . 6  |-  ( y  e.  RR  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  y ) )
1811, 17oveqan12d 6289 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  x.  1 )  +  ( ( _i  x.  y
)  x.  1 ) )  =  ( x  +  ( _i  x.  y ) ) )
1910, 18eqtrd 2495 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( _i  x.  y
) )  x.  1 )  =  ( x  +  ( _i  x.  y ) ) )
20 oveq1 6277 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  ( A  x.  1 )  =  ( ( x  +  ( _i  x.  y ) )  x.  1 ) )
21 id 22 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
2220, 21eqeq12d 2476 . . . 4  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
( A  x.  1 )  =  A  <->  ( (
x  +  ( _i  x.  y ) )  x.  1 )  =  ( x  +  ( _i  x.  y ) ) ) )
2319, 22syl5ibrcom 222 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  ->  ( A  x.  1 )  =  A ) )
2423rexlimivv 2951 . 2  |-  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  ->  ( A  x.  1 )  =  A )
251, 24syl 16 1  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805  (class class class)co 6270   CCcc 9479   RRcr 9480   1c1 9482   _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 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-mulcl 9543  ax-mulcom 9545  ax-mulass 9547  ax-distr 9548  ax-1rid 9551  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  mulid2  9583  mulid1i  9587  mulid1d  9602  muleqadd  10189  divdiv1  10251  conjmul  10257  nnmulcl  10554  expmul  12196  binom21  12269  sq01  12273  bernneq  12277  hashiun  13721  fprodcvg  13822  prodmolem2a  13826  efexp  13921  cncrng  18637  cnfld1  18641  0dgr  22811  ecxp  23225  dvcxp1  23287  efrlim  23500  lgsdilem2  23807  axcontlem7  24478  gxnn0mul  25480  cnrngo  25606  ipasslem2  25948  addltmulALT  27566  zrhnm  28187  dvcncxp1  30343  2even  33012
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