Theorem binom2sub 10589

Description: Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)

Assertion

Ref	Expression
binom2sub	|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )

Proof of Theorem binom2sub

Step	Hyp	Ref	Expression
1		negcl 8119	. . . 4 |- ( B e. CC -> -u B e. CC )
2		binom2 10587	. . . 4 |- ( ( A e. CC /\ -u B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) )
3	1, 2	sylan2 284	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) )
4		negsub 8167	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
5	4	oveq1d 5868	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( A - B ) ^ 2 ) )
6	3, 5	eqtr3d 2205	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) = ( ( A - B ) ^ 2 ) )
7		mulneg2 8315	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. -u B ) = -u ( A x. B ) )
8	7	oveq2d 5869	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. -u B ) ) = ( 2 x. -u ( A x. B ) ) )
9		2cn 8949	. . . . . . 7 |- 2 e. CC
10		mulcl 7901	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
11		mulneg2 8315	. . . . . . 7 |- ( ( 2 e. CC /\ ( A x. B ) e. CC ) -> ( 2 x. -u ( A x. B ) ) = -u ( 2 x. ( A x. B ) ) )
12	9, 10, 11	sylancr 412	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. -u ( A x. B ) ) = -u ( 2 x. ( A x. B ) ) )
13	8, 12	eqtr2d 2204	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> -u ( 2 x. ( A x. B ) ) = ( 2 x. ( A x. -u B ) ) )
14	13	oveq2d 5869	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + -u ( 2 x. ( A x. B ) ) ) = ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) )
15		sqcl 10537	. . . . . 6 |- ( A e. CC -> ( A ^ 2 ) e. CC )
16	15	adantr 274	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) e. CC )
17		mulcl 7901	. . . . . 6 |- ( ( 2 e. CC /\ ( A x. B ) e. CC ) -> ( 2 x. ( A x. B ) ) e. CC )
18	9, 10, 17	sylancr 412	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) e. CC )
19	16, 18	negsubd 8236	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + -u ( 2 x. ( A x. B ) ) ) = ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) )
20	14, 19	eqtr3d 2205	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) = ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) )
21		sqneg 10535	. . . 4 |- ( B e. CC -> ( -u B ^ 2 ) = ( B ^ 2 ) )
22	21	adantl 275	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( -u B ^ 2 ) = ( B ^ 2 ) )
23	20, 22	oveq12d 5871	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
24	6, 23	eqtr3d 2205	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 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   + caddc 7777   x. cmul 7779   - cmin 8090   -ucneg 8091   2c2 8929   ^cexp 10475

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476

This theorem is referenced by:  binom2sub1  10590  binom2subi  10591  resqrexlemover  10974  resqrexlemcalc1  10978  amgm2  11082  bdtrilem  11202  pythagtriplem1  12219  pythagtriplem14  12231  tangtx  13553