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Theorem 1p1times 7379
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 7201 . . . 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 7276 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
5 mulid2 7249 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65, 5oveq12d 5582 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
74, 6eqtrd 2115 1  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434  (class class class)co 5564   CCcc 7111   1c1 7114    + caddc 7116    x. cmul 7118
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 2065  ax-resscn 7200  ax-1cn 7201  ax-icn 7203  ax-addcl 7204  ax-mulcl 7206  ax-mulcom 7209  ax-mulass 7211  ax-distr 7212  ax-1rid 7215  ax-cnre 7219
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5567
This theorem is referenced by:  eqneg  7957  2times  8297
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