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Theorem 3timesi 24344
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 9685 . . 3  |-  3  =  ( 2  +  1 )
21oveq1i 5720 . 2  |-  ( 3  x.  A )  =  ( ( 2  +  1 )  x.  A
)
3 2cn 9696 . . . 4  |-  2  e.  CC
4 ax-1cn 8675 . . . 4  |-  1  e.  CC
5 3timesi.1 . . . 4  |-  A  e.  CC
63, 4, 5adddiri 8728 . . 3  |-  ( ( 2  +  1 )  x.  A )  =  ( ( 2  x.  A )  +  ( 1  x.  A ) )
752timesi 9724 . . . . 5  |-  ( 2  x.  A )  =  ( A  +  A
)
85mulid2i 8720 . . . . 5  |-  ( 1  x.  A )  =  A
97, 8oveq12i 5722 . . . 4  |-  ( ( 2  x.  A )  +  ( 1  x.  A ) )  =  ( ( A  +  A )  +  A
)
105, 5, 5addassi 8725 . . . 4  |-  ( ( A  +  A )  +  A )  =  ( A  +  ( A  +  A ) )
119, 10eqtri 2273 . . 3  |-  ( ( 2  x.  A )  +  ( 1  x.  A ) )  =  ( A  +  ( A  +  A ) )
126, 11eqtri 2273 . 2  |-  ( ( 2  +  1 )  x.  A )  =  ( A  +  ( A  +  A ) )
132, 12eqtri 2273 1  |-  ( 3  x.  A )  =  ( A  +  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621  (class class class)co 5710   CCcc 8615   1c1 8618    + caddc 8620    x. cmul 8622   2c2 9675   3c3 9676
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 1926  ax-ext 2234  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rrecex 8689  ax-cnre 8690
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ov 5713  df-2 9684  df-3 9685
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