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Theorem mulid1 7875
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 7874 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 recn 7865 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
3 ax-icn 7827 . . . . . . 7  |-  _i  e.  CC
4 recn 7865 . . . . . . 7  |-  ( y  e.  RR  ->  y  e.  CC )
5 mulcl 7859 . . . . . . 7  |-  ( ( _i  e.  CC  /\  y  e.  CC )  ->  ( _i  x.  y
)  e.  CC )
63, 4, 5sylancr 411 . . . . . 6  |-  ( y  e.  RR  ->  (
_i  x.  y )  e.  CC )
7 ax-1cn 7825 . . . . . . 7  |-  1  e.  CC
8 adddir 7869 . . . . . . 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 1308 . . . . . 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 287 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( _i  x.  y
) )  x.  1 )  =  ( ( x  x.  1 )  +  ( ( _i  x.  y )  x.  1 ) ) )
11 ax-1rid 7839 . . . . . 6  |-  ( x  e.  RR  ->  (
x  x.  1 )  =  x )
12 mulass 7863 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  y  e.  CC  /\  1  e.  CC )  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  ( y  x.  1 ) ) )
133, 7, 12mp3an13 1310 . . . . . . . 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 7839 . . . . . . . 8  |-  ( y  e.  RR  ->  (
y  x.  1 )  =  y )
1615oveq2d 5840 . . . . . . 7  |-  ( y  e.  RR  ->  (
_i  x.  ( y  x.  1 ) )  =  ( _i  x.  y
) )
1714, 16eqtrd 2190 . . . . . 6  |-  ( y  e.  RR  ->  (
( _i  x.  y
)  x.  1 )  =  ( _i  x.  y ) )
1811, 17oveqan12d 5843 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  x.  1 )  +  ( ( _i  x.  y
)  x.  1 ) )  =  ( x  +  ( _i  x.  y ) ) )
1910, 18eqtrd 2190 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( _i  x.  y
) )  x.  1 )  =  ( x  +  ( _i  x.  y ) ) )
20 oveq1 5831 . . . . 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 2172 . . . 4  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
( A  x.  1 )  =  A  <->  ( (
x  +  ( _i  x.  y ) )  x.  1 )  =  ( x  +  ( _i  x.  y ) ) ) )
2319, 22syl5ibrcom 156 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  ->  ( A  x.  1 )  =  A ) )
2423rexlimivv 2580 . 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 103    = wceq 1335    e. wcel 2128   E.wrex 2436  (class class class)co 5824   CCcc 7730   RRcr 7731   1c1 7733   _ici 7734    + caddc 7735    x. cmul 7737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-mulcl 7830  ax-mulcom 7833  ax-mulass 7835  ax-distr 7836  ax-1rid 7839  ax-cnre 7843
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5135  df-fv 5178  df-ov 5827
This theorem is referenced by:  mulid2  7876  mulid1i  7880  mulid1d  7895  muleqadd  8542  divdivap1  8596  conjmulap  8602  nnmulcl  8854  expmul  10464  binom21  10530  binom2sub1  10532  bernneq  10538  hashiun  11375  fproddccvg  11469  prodmodclem2a  11473  efexp  11579  ecxp  13233
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