Theorem binom2i 10553

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 7894	. . . 4 |- ( A + B ) e. CC
4	3, 1, 2	adddii 7900	. . 3 |- ( ( A + B ) x. ( A + B ) ) = ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) )
5	1, 2, 1	adddiri 7901	. . . . . 6 |- ( ( A + B ) x. A ) = ( ( A x. A ) + ( B x. A ) )
6	2, 1	mulcomi 7896	. . . . . . 7 |- ( B x. A ) = ( A x. B )
7	6	oveq2i 5847	. . . . . 6 |- ( ( A x. A ) + ( B x. A ) ) = ( ( A x. A ) + ( A x. B ) )
8	5, 7	eqtri 2185	. . . . 5 |- ( ( A + B ) x. A ) = ( ( A x. A ) + ( A x. B ) )
9	1, 2, 2	adddiri 7901	. . . . 5 |- ( ( A + B ) x. B ) = ( ( A x. B ) + ( B x. B ) )
10	8, 9	oveq12i 5848	. . . 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 7895	. . . . . 6 |- ( A x. A ) e. CC
12	1, 2	mulcli 7895	. . . . . 6 |- ( A x. B ) e. CC
13	11, 12	addcli 7894	. . . . 5 |- ( ( A x. A ) + ( A x. B ) ) e. CC
14	2, 2	mulcli 7895	. . . . 5 |- ( B x. B ) e. CC
15	13, 12, 14	addassi 7898	. . . 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 7898	. . . . 5 |- ( ( ( A x. A ) + ( A x. B ) ) + ( A x. B ) ) = ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) )
17	16	oveq1i 5846	. . . 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 2191	. . 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 2185	. 2 |- ( ( A + B ) x. ( A + B ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) )
20	3	sqvali 10524	. 2 |- ( ( A + B ) ^ 2 ) = ( ( A + B ) x. ( A + B ) )
21	1	sqvali 10524	. . . 4 |- ( A ^ 2 ) = ( A x. A )
22	12	2timesi 8978	. . . 4 |- ( 2 x. ( A x. B ) ) = ( ( A x. B ) + ( A x. B ) )
23	21, 22	oveq12i 5848	. . 3 |- ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) = ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) )
24	2	sqvali 10524	. . 3 |- ( B ^ 2 ) = ( B x. B )
25	23, 24	oveq12i 5848	. 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 2195	1 |- ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) )

Syntax hints:   = wceq 1342   e. wcel 2135  (class class class)co 5836   CCcc 7742   + caddc 7747   x. cmul 7749   2c2 8899   ^cexp 10444

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-n0 9106  df-z 9183  df-uz 9458  df-seqfrec 10371  df-exp 10445

This theorem is referenced by: (None)