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Theorem 1p1times 7679
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 7501 . . . 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 7576 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
5 mulid2 7549 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65, 5oveq12d 5686 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
74, 6eqtrd 2121 1  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    e. wcel 1439  (class class class)co 5668   CCcc 7411   1c1 7414    + caddc 7416    x. cmul 7418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-resscn 7500  ax-1cn 7501  ax-icn 7503  ax-addcl 7504  ax-mulcl 7506  ax-mulcom 7509  ax-mulass 7511  ax-distr 7512  ax-1rid 7515  ax-cnre 7519
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-iota 4995  df-fv 5038  df-ov 5671
This theorem is referenced by:  eqneg  8262  2times  8607
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