Theorem binom2 10873

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 8124	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
2		simpl 109	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
3		simpr 110	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
4	1, 2, 3	adddid 8171	. . 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 8172	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( B x. A ) ) )
6	3, 2	mulcomd 8168	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( B x. A ) = ( A x. B ) )
7	6	oveq2d 6017	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( B x. A ) ) = ( ( A x. A ) + ( A x. B ) ) )
8	5, 7	eqtrd 2262	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( A x. B ) ) )
9	2, 3, 3	adddird 8172	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. B ) = ( ( A x. B ) + ( B x. B ) ) )
10	8, 9	oveq12d 6019	. . . 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 8167	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. A ) e. CC )
12	2, 3	mulcld 8167	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13	11, 12	addcld 8166	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( A x. B ) ) e. CC )
14	3, 3	mulcld 8167	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
15	13, 12, 14	addassd 8169	. . . 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 8169	. . . . 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 6016	. . . 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 2268	. . 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 2262	. 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 10819	. . 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 10819	. . . . 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 9354	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) = ( ( A x. B ) + ( A x. B ) ) )
25	23, 24	oveq12d 6019	. . 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 10819	. . . 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 6019	. 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 2272	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200  (class class class)co 6001   CCcc 7997   + caddc 8002   x. cmul 8004   2c2 9161   ^cexp 10760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-seqfrec 10670  df-exp 10761

This theorem is referenced by:  binom21  10874  binom2sub  10875  mulbinom2  10878  binom3  10879  nn0opthlem1d  10942  resqrexlemover  11521  resqrexlemcalc1  11525  abstri  11615  amgm2  11629  bdtrilem  11750  pythagtriplem1  12788  pythagtriplem12  12798