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Theorem mulid1 7178
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 7177 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 recn 7168 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
3 ax-icn 7133 . . . . . . 7  |-  _i  e.  CC
4 recn 7168 . . . . . . 7  |-  ( y  e.  RR  ->  y  e.  CC )
5 mulcl 7162 . . . . . . 7  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
63, 4, 5sylancr 405 . . . . . 6  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
7 ax-1cn 7131 . . . . . . 7  |-  1  e.  CC
8 adddir 7172 . . . . . . 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 1258 . . . . . 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 283 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( _i  x.  y
) )  x.  1 )  =  ( ( x  x.  1 )  +  ( ( _i  x.  y )  x.  1 ) ) )
11 ax-1rid 7145 . . . . . 6  |-  ( x  e.  RR  ->  (
x  x.  1 )  =  x )
12 mulass 7166 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  y  e.  CC  /\  1  e.  CC )  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  ( y  x.  1 ) ) )
133, 7, 12mp3an13 1260 . . . . . . . 8  |-  ( y  e.  CC  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  ( y  x.  1 ) ) )
144, 13syl 14 . . . . . . 7  |-  ( y  e.  RR  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  ( y  x.  1 ) ) )
15 ax-1rid 7145 . . . . . . . 8  |-  ( y  e.  RR  ->  (
y  x.  1 )  =  y )
1615oveq2d 5559 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  ( y  x.  1 ) )  =  ( _i  x.  y
) )
1714, 16eqtrd 2114 . . . . . 6  |-  ( y  e.  RR  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  y ) )
1811, 17oveqan12d 5562 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  x.  1 )  +  ( ( _i  x.  y
)  x.  1 ) )  =  ( x  +  ( _i  x.  y ) ) )
1910, 18eqtrd 2114 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( _i  x.  y
) )  x.  1 )  =  ( x  +  ( _i  x.  y ) ) )
20 oveq1 5550 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  ( A  x.  1 )  =  ( ( x  +  ( _i  x.  y ) )  x.  1 ) )
21 id 19 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
2220, 21eqeq12d 2096 . . . 4  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
( A  x.  1 )  =  A  <->  ( (
x  +  ( _i  x.  y ) )  x.  1 )  =  ( x  +  ( _i  x.  y ) ) ) )
2319, 22syl5ibrcom 155 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  ->  ( A  x.  1 )  =  A ) )
2423rexlimivv 2483 . 2  |-  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  ->  ( A  x.  1 )  =  A )
251, 24syl 14 1  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   E.wrex 2350  (class class class)co 5543   CCcc 7041   RRcr 7042   1c1 7044   _ici 7045    + caddc 7046    x. cmul 7048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-mulcl 7136  ax-mulcom 7139  ax-mulass 7141  ax-distr 7142  ax-1rid 7145  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  mulid2  7179  mulid1i  7183  mulid1d  7198  muleqadd  7825  divdivap1  7878  conjmulap  7884  nnmulcl  8127  expmul  9618  binom21  9683  bernneq  9690
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