Theorem binom2i 10401

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 7770	. . . 4 |- ( A + B ) e. CC
4	3, 1, 2	adddii 7776	. . 3 |- ( ( A + B ) x. ( A + B ) ) = ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) )
5	1, 2, 1	adddiri 7777	. . . . . 6 |- ( ( A + B ) x. A ) = ( ( A x. A ) + ( B x. A ) )
6	2, 1	mulcomi 7772	. . . . . . 7 |- ( B x. A ) = ( A x. B )
7	6	oveq2i 5785	. . . . . 6 |- ( ( A x. A ) + ( B x. A ) ) = ( ( A x. A ) + ( A x. B ) )
8	5, 7	eqtri 2160	. . . . 5 |- ( ( A + B ) x. A ) = ( ( A x. A ) + ( A x. B ) )
9	1, 2, 2	adddiri 7777	. . . . 5 |- ( ( A + B ) x. B ) = ( ( A x. B ) + ( B x. B ) )
10	8, 9	oveq12i 5786	. . . 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 7771	. . . . . 6 |- ( A x. A ) e. CC
12	1, 2	mulcli 7771	. . . . . 6 |- ( A x. B ) e. CC
13	11, 12	addcli 7770	. . . . 5 |- ( ( A x. A ) + ( A x. B ) ) e. CC
14	2, 2	mulcli 7771	. . . . 5 |- ( B x. B ) e. CC
15	13, 12, 14	addassi 7774	. . . 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 7774	. . . . 5 |- ( ( ( A x. A ) + ( A x. B ) ) + ( A x. B ) ) = ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) )
17	16	oveq1i 5784	. . . 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 2166	. . 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 2160	. 2 |- ( ( A + B ) x. ( A + B ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) )
20	3	sqvali 10372	. 2 |- ( ( A + B ) ^ 2 ) = ( ( A + B ) x. ( A + B ) )
21	1	sqvali 10372	. . . 4 |- ( A ^ 2 ) = ( A x. A )
22	12	2timesi 8850	. . . 4 |- ( 2 x. ( A x. B ) ) = ( ( A x. B ) + ( A x. B ) )
23	21, 22	oveq12i 5786	. . 3 |- ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) = ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) )
24	2	sqvali 10372	. . 3 |- ( B ^ 2 ) = ( B x. B )
25	23, 24	oveq12i 5786	. 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 2170	1 |- ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) )

Syntax hints:   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   + caddc 7623   x. cmul 7625   2c2 8771   ^cexp 10292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327  df-seqfrec 10219  df-exp 10293

This theorem is referenced by: (None)