Theorem binom2sub 10619

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 8147	. . . 4 |- ( B e. CC -> -u B e. CC )
2		binom2 10617	. . . 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 286	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) )
4		negsub 8195	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
5	4	oveq1d 5884	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( A - B ) ^ 2 ) )
6	3, 5	eqtr3d 2212	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) = ( ( A - B ) ^ 2 ) )
7		mulneg2 8343	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. -u B ) = -u ( A x. B ) )
8	7	oveq2d 5885	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. -u B ) ) = ( 2 x. -u ( A x. B ) ) )
9		2cn 8979	. . . . . . 7 |- 2 e. CC
10		mulcl 7929	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
11		mulneg2 8343	. . . . . . 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 414	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. -u ( A x. B ) ) = -u ( 2 x. ( A x. B ) ) )
13	8, 12	eqtr2d 2211	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> -u ( 2 x. ( A x. B ) ) = ( 2 x. ( A x. -u B ) ) )
14	13	oveq2d 5885	. . . 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 10567	. . . . . 6 |- ( A e. CC -> ( A ^ 2 ) e. CC )
16	15	adantr 276	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) e. CC )
17		mulcl 7929	. . . . . 6 |- ( ( 2 e. CC /\ ( A x. B ) e. CC ) -> ( 2 x. ( A x. B ) ) e. CC )
18	9, 10, 17	sylancr 414	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) e. CC )
19	16, 18	negsubd 8264	. . . 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 2212	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) = ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) )
21		sqneg 10565	. . . 4 |- ( B e. CC -> ( -u B ^ 2 ) = ( B ^ 2 ) )
22	21	adantl 277	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( -u B ^ 2 ) = ( B ^ 2 ) )
23	20, 22	oveq12d 5887	. 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 2212	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 1353   e. wcel 2148  (class class class)co 5869   CCcc 7800   + caddc 7805   x. cmul 7807   - cmin 8118   -ucneg 8119   2c2 8959   ^cexp 10505

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-seqfrec 10432  df-exp 10506

This theorem is referenced by:  binom2sub1  10620  binom2subi  10621  resqrexlemover  11003  resqrexlemcalc1  11007  amgm2  11111  bdtrilem  11231  pythagtriplem1  12248  pythagtriplem14  12260  tangtx  13926