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Theorem 1p1times 8983
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 8795 . . . 4  |-  1  e.  CC
21a1i 10 . . 3  |-  ( A  e.  CC  ->  1  e.  CC )
3 id 19 . . 3  |-  ( A  e.  CC  ->  A  e.  CC )
42, 2, 3adddird 8860 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
5 mulid2 8836 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65, 5oveq12d 5876 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
74, 6eqtrd 2315 1  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742
This theorem is referenced by:  addcom  8998  addcomd  9014  eqneg  9480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-mulcom 8801  ax-mulass 8803  ax-distr 8804  ax-1rid 8807  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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