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Theorem 3timesi 24590
Description: Three times a number. (Contributed by FL, 17-Oct-2010.)
Hypothesis
Ref Expression
3timesi.1  |-  A  e.  CC
Assertion
Ref Expression
3timesi  |-  ( 3  x.  A )  =  ( A  +  ( A  +  A ) )

Proof of Theorem 3timesi
StepHypRef Expression
1 df-3 9805 . . 3  |-  3  =  ( 2  +  1 )
21oveq1i 5868 . 2  |-  ( 3  x.  A )  =  ( ( 2  +  1 )  x.  A
)
3 2cn 9816 . . . 4  |-  2  e.  CC
4 ax-1cn 8795 . . . 4  |-  1  e.  CC
5 3timesi.1 . . . 4  |-  A  e.  CC
63, 4, 5adddiri 8848 . . 3  |-  ( ( 2  +  1 )  x.  A )  =  ( ( 2  x.  A )  +  ( 1  x.  A ) )
752timesi 9845 . . . . 5  |-  ( 2  x.  A )  =  ( A  +  A
)
85mulid2i 8840 . . . . 5  |-  ( 1  x.  A )  =  A
97, 8oveq12i 5870 . . . 4  |-  ( ( 2  x.  A )  +  ( 1  x.  A ) )  =  ( ( A  +  A )  +  A
)
105, 5, 5addassi 8845 . . . 4  |-  ( ( A  +  A )  +  A )  =  ( A  +  ( A  +  A ) )
119, 10eqtri 2303 . . 3  |-  ( ( 2  x.  A )  +  ( 1  x.  A ) )  =  ( A  +  ( A  +  A ) )
126, 11eqtri 2303 . 2  |-  ( ( 2  +  1 )  x.  A )  =  ( A  +  ( A  +  A ) )
132, 12eqtri 2303 1  |-  ( 3  x.  A )  =  ( A  +  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   2c2 9795   3c3 9796
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-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rrecex 8809  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-ne 2448  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  df-2 9804  df-3 9805
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