Theorem binom2 10562

Description: The square of a binomial. (Contributed by FL, 10-Dec-2006.)

Assertion

Ref	Expression
binom2	|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )

Proof of Theorem binom2

Step	Hyp	Ref	Expression
1		addcl 7874	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
2		simpl 108	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
3		simpr 109	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
4	1, 2, 3	adddid 7919	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. ( A + B ) ) = ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) ) )
5	2, 3, 2	adddird 7920	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( B x. A ) ) )
6	3, 2	mulcomd 7916	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( B x. A ) = ( A x. B ) )
7	6	oveq2d 5857	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( B x. A ) ) = ( ( A x. A ) + ( A x. B ) ) )
8	5, 7	eqtrd 2198	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( A x. B ) ) )
9	2, 3, 3	adddird 7920	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. B ) = ( ( A x. B ) + ( B x. B ) ) )
10	8, 9	oveq12d 5859	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) ) = ( ( ( A x. A ) + ( A x. B ) ) + ( ( A x. B ) + ( B x. B ) ) ) )
11	2, 2	mulcld 7915	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. A ) e. CC )
12	2, 3	mulcld 7915	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13	11, 12	addcld 7914	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( A x. B ) ) e. CC )
14	3, 3	mulcld 7915	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
15	13, 12, 14	addassd 7917	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( 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	addassd 7917	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. A ) + ( A x. B ) ) + ( A x. B ) ) = ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) )
17	16	oveq1d 5856	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( 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	3eqtr2d 2204	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) x. A ) + ( ( A + B ) x. B ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) ) )
19	4, 18	eqtrd 2198	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. ( A + B ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) ) )
20		sqval 10509	. . 3 |- ( ( A + B ) e. CC -> ( ( A + B ) ^ 2 ) = ( ( A + B ) x. ( A + B ) ) )
21	1, 20	syl 14	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) = ( ( A + B ) x. ( A + B ) ) )
22		sqval 10509	. . . . 5 |- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
23	2, 22	syl 14	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) = ( A x. A ) )
24	12	2timesd 9095	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) = ( ( A x. B ) + ( A x. B ) ) )
25	23, 24	oveq12d 5859	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) = ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) )
26		sqval 10509	. . . 4 |- ( B e. CC -> ( B ^ 2 ) = ( B x. B ) )
27	3, 26	syl 14	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) = ( B x. B ) )
28	25, 27	oveq12d 5859	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) = ( ( ( A x. A ) + ( ( A x. B ) + ( A x. B ) ) ) + ( B x. B ) ) )
29	19, 21, 28	3eqtr4d 2208	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136  (class class class)co 5841   CCcc 7747   + caddc 7752   x. cmul 7754   2c2 8904   ^cexp 10450

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-n0 9111  df-z 9188  df-uz 9463  df-seqfrec 10377  df-exp 10451

This theorem is referenced by:  binom21  10563  binom2sub  10564  mulbinom2  10567  binom3  10568  nn0opthlem1d  10629  resqrexlemover  10948  resqrexlemcalc1  10952  abstri  11042  amgm2  11056  bdtrilem  11176  pythagtriplem1  12193  pythagtriplem12  12203