Theorem subsq2 10756

Description: Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)

Assertion

Ref	Expression
subsq2	|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( ( A - B ) ^ 2 ) + ( ( 2 x. B ) x. ( A - B ) ) ) )

Proof of Theorem subsq2

Step	Hyp	Ref	Expression
1		2cn 9078	. . . . . . . 8 |- 2 e. CC
2		mulcl 8023	. . . . . . . 8 |- ( ( 2 e. CC /\ B e. CC ) -> ( 2 x. B ) e. CC )
3	1, 2	mpan 424	. . . . . . 7 |- ( B e. CC -> ( 2 x. B ) e. CC )
4	3	adantl 277	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. B ) e. CC )
5		subadd23 8255	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ ( 2 x. B ) e. CC ) -> ( ( A - B ) + ( 2 x. B ) ) = ( A + ( ( 2 x. B ) - B ) ) )
6	4, 5	mpd3an3 1349	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + ( 2 x. B ) ) = ( A + ( ( 2 x. B ) - B ) ) )
7		2times 9135	. . . . . . . . 9 |- ( B e. CC -> ( 2 x. B ) = ( B + B ) )
8	7	oveq1d 5940	. . . . . . . 8 |- ( B e. CC -> ( ( 2 x. B ) - B ) = ( ( B + B ) - B ) )
9		pncan 8249	. . . . . . . . 9 |- ( ( B e. CC /\ B e. CC ) -> ( ( B + B ) - B ) = B )
10	9	anidms 397	. . . . . . . 8 |- ( B e. CC -> ( ( B + B ) - B ) = B )
11	8, 10	eqtrd 2229	. . . . . . 7 |- ( B e. CC -> ( ( 2 x. B ) - B ) = B )
12	11	adantl 277	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( 2 x. B ) - B ) = B )
13	12	oveq2d 5941	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( ( 2 x. B ) - B ) ) = ( A + B ) )
14	6, 13	eqtrd 2229	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + ( 2 x. B ) ) = ( A + B ) )
15	14	oveq1d 5940	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - B ) + ( 2 x. B ) ) x. ( A - B ) ) = ( ( A + B ) x. ( A - B ) ) )
16		subcl 8242	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
17	16, 4, 16	adddird 8069	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - B ) + ( 2 x. B ) ) x. ( A - B ) ) = ( ( ( A - B ) x. ( A - B ) ) + ( ( 2 x. B ) x. ( A - B ) ) ) )
18	15, 17	eqtr3d 2231	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. ( A - B ) ) = ( ( ( A - B ) x. ( A - B ) ) + ( ( 2 x. B ) x. ( A - B ) ) ) )
19		subsq 10755	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) )
20		sqval 10706	. . . 4 |- ( ( A - B ) e. CC -> ( ( A - B ) ^ 2 ) = ( ( A - B ) x. ( A - B ) ) )
21	16, 20	syl 14	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) ^ 2 ) = ( ( A - B ) x. ( A - B ) ) )
22	21	oveq1d 5940	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - B ) ^ 2 ) + ( ( 2 x. B ) x. ( A - B ) ) ) = ( ( ( A - B ) x. ( A - B ) ) + ( ( 2 x. B ) x. ( A - B ) ) ) )
23	18, 19, 22	3eqtr4d 2239	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( ( A - B ) ^ 2 ) + ( ( 2 x. B ) x. ( A - B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167  (class class class)co 5925   CCcc 7894   + caddc 7899   x. cmul 7901   - cmin 8214   2c2 9058   ^cexp 10647

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-seqfrec 10557  df-exp 10648

This theorem is referenced by: (None)