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Theorem 1p1times 8121
Description: Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
1p1times  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )

Proof of Theorem 1p1times
StepHypRef Expression
1 ax-1cn 7934 . . . 4  |-  1  e.  CC
21a1i 9 . . 3  |-  ( A  e.  CC  ->  1  e.  CC )
3 id 19 . . 3  |-  ( A  e.  CC  ->  A  e.  CC )
42, 2, 3adddird 8013 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
5 mullid 7985 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65, 5oveq12d 5914 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
74, 6eqtrd 2222 1  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160  (class class class)co 5896   CCcc 7839   1c1 7842    + caddc 7844    x. cmul 7846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-resscn 7933  ax-1cn 7934  ax-icn 7936  ax-addcl 7937  ax-mulcl 7939  ax-mulcom 7942  ax-mulass 7944  ax-distr 7945  ax-1rid 7948  ax-cnre 7952
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899
This theorem is referenced by:  eqneg  8719  2times  9077
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