Theorem binom2sub 10540

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 8079	. . . 4 |- ( B e. CC -> -u B e. CC )
2		binom2 10538	. . . 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 8127	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
5	4	oveq1d 5841	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) ^ 2 ) = ( ( A - B ) ^ 2 ) )
6	3, 5	eqtr3d 2192	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) + ( -u B ^ 2 ) ) = ( ( A - B ) ^ 2 ) )
7		mulneg2 8275	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. -u B ) = -u ( A x. B ) )
8	7	oveq2d 5842	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. -u B ) ) = ( 2 x. -u ( A x. B ) ) )
9		2cn 8909	. . . . . . 7 |- 2 e. CC
10		mulcl 7861	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
11		mulneg2 8275	. . . . . . 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 411	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. -u ( A x. B ) ) = -u ( 2 x. ( A x. B ) ) )
13	8, 12	eqtr2d 2191	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> -u ( 2 x. ( A x. B ) ) = ( 2 x. ( A x. -u B ) ) )
14	13	oveq2d 5842	. . . 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 10489	. . . . . 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 7861	. . . . . 6 |- ( ( 2 e. CC /\ ( A x. B ) e. CC ) -> ( 2 x. ( A x. B ) ) e. CC )
18	9, 10, 17	sylancr 411	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) e. CC )
19	16, 18	negsubd 8196	. . . 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 2192	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + ( 2 x. ( A x. -u B ) ) ) = ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) )
21		sqneg 10487	. . . 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 5844	. 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 2192	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 1335   e. wcel 2128  (class class class)co 5826   CCcc 7732   + caddc 7737   x. cmul 7739   - cmin 8050   -ucneg 8051   2c2 8889   ^cexp 10427

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-n0 9096  df-z 9173  df-uz 9445  df-seqfrec 10354  df-exp 10428

This theorem is referenced by:  binom2sub1  10541  binom2subi  10542  resqrexlemover  10921  resqrexlemcalc1  10925  amgm2  11029  bdtrilem  11149  pythagtriplem1  12155  pythagtriplem14  12167  tangtx  13229