Theorem binom2 10725

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 7999	. . . 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 8046	. . 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 8047	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( B x. A ) ) )
6	3, 2	mulcomd 8043	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( B x. A ) = ( A x. B ) )
7	6	oveq2d 5935	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( B x. A ) ) = ( ( A x. A ) + ( A x. B ) ) )
8	5, 7	eqtrd 2226	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( A x. B ) ) )
9	2, 3, 3	adddird 8047	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. B ) = ( ( A x. B ) + ( B x. B ) ) )
10	8, 9	oveq12d 5937	. . . 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 8042	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. A ) e. CC )
12	2, 3	mulcld 8042	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13	11, 12	addcld 8041	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( A x. B ) ) e. CC )
14	3, 3	mulcld 8042	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
15	13, 12, 14	addassd 8044	. . . 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 8044	. . . . 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 5934	. . . 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 2232	. . 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 2226	. 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 10671	. . 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 10671	. . . . 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 9228	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) = ( ( A x. B ) + ( A x. B ) ) )
25	23, 24	oveq12d 5937	. . 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 10671	. . . 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 5937	. 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 2236	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 1364   e. wcel 2164  (class class class)co 5919   CCcc 7872   + caddc 7877   x. cmul 7879   2c2 9035   ^cexp 10612

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522  df-exp 10613

This theorem is referenced by:  binom21  10726  binom2sub  10727  mulbinom2  10730  binom3  10731  nn0opthlem1d  10794  resqrexlemover  11157  resqrexlemcalc1  11161  abstri  11251  amgm2  11265  bdtrilem  11385  pythagtriplem1  12406  pythagtriplem12  12416