Theorem subsq2 10739

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 9061	. . . . . . . 8 |- 2 e. CC
2		mulcl 8006	. . . . . . . 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 8238	. . . . . 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 9118	. . . . . . . . 9 |- ( B e. CC -> ( 2 x. B ) = ( B + B ) )
8	7	oveq1d 5937	. . . . . . . 8 |- ( B e. CC -> ( ( 2 x. B ) - B ) = ( ( B + B ) - B ) )
9		pncan 8232	. . . . . . . . 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 5938	. . . . 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 5937	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - B ) + ( 2 x. B ) ) x. ( A - B ) ) = ( ( A + B ) x. ( A - B ) ) )
16		subcl 8225	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
17	16, 4, 16	adddird 8052	. . 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 10738	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) )
20		sqval 10689	. . . 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 5937	. 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 5922   CCcc 7877   + caddc 7882   x. cmul 7884   - cmin 8197   2c2 9041   ^cexp 10630

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540  df-exp 10631

This theorem is referenced by: (None)