Theorem subsq2 11009

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 9308	. . . . . . . 8 |- 2 e. CC
2		mulcl 8254	. . . . . . . 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 8485	. . . . . 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 1375	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + ( 2 x. B ) ) = ( A + ( ( 2 x. B ) - B ) ) )
7		2times 9365	. . . . . . . . 9 |- ( B e. CC -> ( 2 x. B ) = ( B + B ) )
8	7	oveq1d 6065	. . . . . . . 8 |- ( B e. CC -> ( ( 2 x. B ) - B ) = ( ( B + B ) - B ) )
9		pncan 8479	. . . . . . . . 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 2265	. . . . . . 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 6066	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( ( 2 x. B ) - B ) ) = ( A + B ) )
14	6, 13	eqtrd 2265	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + ( 2 x. B ) ) = ( A + B ) )
15	14	oveq1d 6065	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - B ) + ( 2 x. B ) ) x. ( A - B ) ) = ( ( A + B ) x. ( A - B ) ) )
16		subcl 8472	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
17	16, 4, 16	adddird 8299	. . 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 2267	. 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 11008	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) )
20		sqval 10959	. . . 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 6065	. 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 2275	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 1398   e. wcel 2203  (class class class)co 6050   CCcc 8125   + caddc 8130   x. cmul 8132   - cmin 8444   2c2 9288   ^cexp 10900

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810  df-exp 10901

This theorem is referenced by: (None)