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Theorem mulid2 7083
Description: Identity law for multiplication. Note: see mulid1 7082 for commuted version. (Contributed by NM, 8-Oct-1999.)
Assertion
Ref Expression
mulid2  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )

Proof of Theorem mulid2
StepHypRef Expression
1 ax-1cn 7035 . . 3  |-  1  e.  CC
2 mulcom 7068 . . 3  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  x.  A
)  =  ( A  x.  1 ) )
31, 2mpan 408 . 2  |-  ( A  e.  CC  ->  (
1  x.  A )  =  ( A  x.  1 ) )
4 mulid1 7082 . 2  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
53, 4eqtrd 2088 1  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409  (class class class)co 5540   CCcc 6945   1c1 6948    x. cmul 6952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-mulcl 7040  ax-mulcom 7043  ax-mulass 7045  ax-distr 7046  ax-1rid 7049  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by:  mulid2i  7088  mulid2d  7103  muladd11  7207  1p1times  7208  mulm1  7469  div1  7754  recdivap  7769  divdivap2  7775  conjmulap  7780  expp1  9427  recan  9936
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