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Theorem 3timesi 24577
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 9800 . . 3  |-  3  =  ( 2  +  1 )
21oveq1i 5829 . 2  |-  ( 3  x.  A )  =  ( ( 2  +  1 )  x.  A
)
3 2cn 9811 . . . 4  |-  2  e.  CC
4 ax-1cn 8790 . . . 4  |-  1  e.  CC
5 3timesi.1 . . . 4  |-  A  e.  CC
63, 4, 5adddiri 8843 . . 3  |-  ( ( 2  +  1 )  x.  A )  =  ( ( 2  x.  A )  +  ( 1  x.  A ) )
752timesi 9840 . . . . 5  |-  ( 2  x.  A )  =  ( A  +  A
)
85mulid2i 8835 . . . . 5  |-  ( 1  x.  A )  =  A
97, 8oveq12i 5831 . . . 4  |-  ( ( 2  x.  A )  +  ( 1  x.  A ) )  =  ( ( A  +  A )  +  A
)
105, 5, 5addassi 8840 . . . 4  |-  ( ( A  +  A )  +  A )  =  ( A  +  ( A  +  A ) )
119, 10eqtri 2304 . . 3  |-  ( ( 2  x.  A )  +  ( 1  x.  A ) )  =  ( A  +  ( A  +  A ) )
126, 11eqtri 2304 . 2  |-  ( ( 2  +  1 )  x.  A )  =  ( A  +  ( A  +  A ) )
132, 12eqtri 2304 1  |-  ( 3  x.  A )  =  ( A  +  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1624    e. wcel 1685  (class class class)co 5819   CCcc 8730   1c1 8733    + caddc 8735    x. cmul 8737   2c2 9790   3c3 9791
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rrecex 8804  ax-cnre 8805
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5822  df-2 9799  df-3 9800
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