ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2times Unicode version

Theorem 2times 8999
Description: Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
Assertion
Ref Expression
2times  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )

Proof of Theorem 2times
StepHypRef Expression
1 df-2 8930 . . 3  |-  2  =  ( 1  +  1 )
21oveq1i 5861 . 2  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 1p1times 8046 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
42, 3eqtrid 2215 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141  (class class class)co 5851   CCcc 7765   1c1 7768    + caddc 7770    x. cmul 7772   2c2 8922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7859  ax-1cn 7860  ax-icn 7862  ax-addcl 7863  ax-mulcl 7865  ax-mulcom 7868  ax-mulass 7870  ax-distr 7871  ax-1rid 7874  ax-cnre 7878
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204  df-ov 5854  df-2 8930
This theorem is referenced by:  times2  9000  2timesi  9001  2halves  9100  halfaddsub  9105  avglt2  9110  2timesd  9113  expubnd  10526  subsq2  10576  sinmul  11700  sin2t  11705  cos2t  11706  pythagtriplem4  12215  pythagtriplem14  12224  pythagtriplem16  12226
  Copyright terms: Public domain W3C validator