Theorem binom2i 10242

Description: The square of a binomial. (Contributed by NM, 11-Aug-1999.)

Hypotheses

Ref	Expression
binom2.1	|- A e. CC
binom2.2	|- B e. CC

Assertion

Ref	Expression
binom2i	|- ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) )

Proof of Theorem binom2i

Step	Hyp	Ref	Expression
1		binom2.1	. . . . 5 |- A e. CC
2		binom2.2	. . . . 5 |- B e. CC
3	1, 2	addcli 7642	. . . 4 |- ( A + B ) e. CC
4	3, 1, 2	adddii 7648	. . 3 |- ( ( A + B ) x. ( A + B ) ) = ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) )
5	1, 2, 1	adddiri 7649	. . . . . 6 |- ( ( A + B ) x. A ) = ( ( A x. A ) + ( B x. A ) )
6	2, 1	mulcomi 7644	. . . . . . 7 |- ( B x. A ) = ( A x. B )
7	6	oveq2i 5717	. . . . . 6 |- ( ( A x. A ) + ( B x. A ) ) = ( ( A x. A ) + ( A x. B ) )
8	5, 7	eqtri 2120	. . . . 5 |- ( ( A + B ) x. A ) = ( ( A x. A ) + ( A x. B ) )
9	1, 2, 2	adddiri 7649	. . . . 5 |- ( ( A + B ) x. B ) = ( ( A x. B ) + ( B x. B ) )
10	8, 9	oveq12i 5718	. . . 4 |- ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) ) = ( ( ( A x. A ) + ( A x. B ) ) + ( ( A x. B ) + ( B x. B ) ) )
11	1, 1	mulcli 7643	. . . . . 6 |- ( A x. A ) e. CC
12	1, 2	mulcli 7643	. . . . . 6 |- ( A x. B ) e. CC
13	11, 12	addcli 7642	. . . . 5 |- ( ( A x. A ) + ( A x. B ) ) e. CC
14	2, 2	mulcli 7643	. . . . 5 |- ( B x. B ) e. CC
15	13, 12, 14	addassi 7646	. . . 4 |- ( ( ( ( A x. A ) + ( A x. B ) ) + ( A x. B ) ) + ( B x. B ) ) = ( ( ( A x. A ) + ( A x. B ) ) + ( ( A x. B ) + ( B x. B ) ) )
16	11, 12, 12	addassi 7646	. . . . 5 |- ( ( ( A x. A ) + ( A x. B ) ) + ( A x. B ) ) = ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) )
17	16	oveq1i 5716	. . . 4 |- ( ( ( ( A x. A ) + ( A x. B ) ) + ( A x. B ) ) + ( B x. B ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) )
18	10, 15, 17	3eqtr2i 2126	. . 3 |- ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) )
19	4, 18	eqtri 2120	. 2 |- ( ( A + B ) x. ( A + B ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) )
20	3	sqvali 10213	. 2 |- ( ( A + B ) ^ 2 ) = ( ( A + B ) x. ( A + B ) )
21	1	sqvali 10213	. . . 4 |- ( A ^ 2 ) = ( A x. A )
22	12	2timesi 8702	. . . 4 |- ( 2 x. ( A x. B ) ) = ( ( A x. B ) + ( A x. B ) )
23	21, 22	oveq12i 5718	. . 3 |- ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) = ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) )
24	2	sqvali 10213	. . 3 |- ( B ^ 2 ) = ( B x. B )
25	23, 24	oveq12i 5718	. 2 |- ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) )
26	19, 20, 25	3eqtr4i 2130	1 |- ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) )

Syntax hints:   = wceq 1299   e. wcel 1448  (class class class)co 5706   CCcc 7498   + caddc 7503   x. cmul 7505   2c2 8629   ^cexp 10133

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-n0 8830  df-z 8907  df-uz 9177  df-seqfrec 10060  df-exp 10134

This theorem is referenced by: (None)