Theorem binom2 11037

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 8268	. . . 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 8314	. . 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 8315	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( B x. A ) ) )
6	3, 2	mulcomd 8311	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( B x. A ) = ( A x. B ) )
7	6	oveq2d 6074	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( B x. A ) ) = ( ( A x. A ) + ( A x. B ) ) )
8	5, 7	eqtrd 2267	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. A ) = ( ( A x. A ) + ( A x. B ) ) )
9	2, 3, 3	adddird 8315	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. B ) = ( ( A x. B ) + ( B x. B ) ) )
10	8, 9	oveq12d 6076	. . . 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 8310	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. A ) e. CC )
12	2, 3	mulcld 8310	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13	11, 12	addcld 8309	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) + ( A x. B ) ) e. CC )
14	3, 3	mulcld 8310	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
15	13, 12, 14	addassd 8312	. . . 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 8312	. . . . 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 6073	. . . 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 2273	. . 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 2267	. 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 10983	. . 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 10983	. . . . 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 9498	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) = ( ( A x. B ) + ( A x. B ) ) )
25	23, 24	oveq12d 6076	. . 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 10983	. . . 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 6076	. 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 2277	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 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   + caddc 8146   x. cmul 8148   2c2 9305   ^cexp 10924

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  binom21  11038  binom2sub  11039  mulbinom2  11042  binom3  11043  nn0opthlem1d  11107  resqrexlemover  11720  resqrexlemcalc1  11724  abstri  11814  amgm2  11828  bdtrilem  11949  pythagtriplem1  12988  pythagtriplem12  12998