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Theorem 3timesi 24531
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 9759 . . 3  |-  3  =  ( 2  +  1 )
21oveq1i 5788 . 2  |-  ( 3  x.  A )  =  ( ( 2  +  1 )  x.  A
)
3 2cn 9770 . . . 4  |-  2  e.  CC
4 ax-1cn 8749 . . . 4  |-  1  e.  CC
5 3timesi.1 . . . 4  |-  A  e.  CC
63, 4, 5adddiri 8802 . . 3  |-  ( ( 2  +  1 )  x.  A )  =  ( ( 2  x.  A )  +  ( 1  x.  A ) )
752timesi 9798 . . . . 5  |-  ( 2  x.  A )  =  ( A  +  A
)
85mulid2i 8794 . . . . 5  |-  ( 1  x.  A )  =  A
97, 8oveq12i 5790 . . . 4  |-  ( ( 2  x.  A )  +  ( 1  x.  A ) )  =  ( ( A  +  A )  +  A
)
105, 5, 5addassi 8799 . . . 4  |-  ( ( A  +  A )  +  A )  =  ( A  +  ( A  +  A ) )
119, 10eqtri 2276 . . 3  |-  ( ( 2  x.  A )  +  ( 1  x.  A ) )  =  ( A  +  ( A  +  A ) )
126, 11eqtri 2276 . 2  |-  ( ( 2  +  1 )  x.  A )  =  ( A  +  ( A  +  A ) )
132, 12eqtri 2276 1  |-  ( 3  x.  A )  =  ( A  +  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621  (class class class)co 5778   CCcc 8689   1c1 8692    + caddc 8694    x. cmul 8696   2c2 9749   3c3 9750
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rrecex 8763  ax-cnre 8764
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fv 4675  df-ov 5781  df-2 9758  df-3 9759
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