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Theorem 1p1times 8999
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 8811 . . . 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 8876 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
5 mulid2 8852 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
65, 5oveq12d 5892 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
74, 6eqtrd 2328 1  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758
This theorem is referenced by:  addcom  9014  addcomd  9030  eqneg  9496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-mulcom 8817  ax-mulass 8819  ax-distr 8820  ax-1rid 8823  ax-cnre 8826
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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