Theorem subsq 10582

Description: Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)

Assertion

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

Proof of Theorem subsq

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2		simpr 109	. . 3 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
3		subcl 8118	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
4	1, 2, 3	adddird 7945	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. ( A - B ) ) = ( ( A x. ( A - B ) ) + ( B x. ( A - B ) ) ) )
5		subdi 8304	. . . . 5 |- ( ( A e. CC /\ A e. CC /\ B e. CC ) -> ( A x. ( A - B ) ) = ( ( A x. A ) - ( A x. B ) ) )
6	5	3anidm12 1290	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( A - B ) ) = ( ( A x. A ) - ( A x. B ) ) )
7		sqval 10534	. . . . . 6 |- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
8	7	adantr 274	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) = ( A x. A ) )
9	8	oveq1d 5868	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( A x. B ) ) = ( ( A x. A ) - ( A x. B ) ) )
10	6, 9	eqtr4d 2206	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( A - B ) ) = ( ( A ^ 2 ) - ( A x. B ) ) )
11	2, 1, 2	subdid 8333	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B x. ( A - B ) ) = ( ( B x. A ) - ( B x. B ) ) )
12		mulcom 7903	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
13		sqval 10534	. . . . . 6 |- ( B e. CC -> ( B ^ 2 ) = ( B x. B ) )
14	13	adantl 275	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) = ( B x. B ) )
15	12, 14	oveq12d 5871	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) - ( B ^ 2 ) ) = ( ( B x. A ) - ( B x. B ) ) )
16	11, 15	eqtr4d 2206	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B x. ( A - B ) ) = ( ( A x. B ) - ( B ^ 2 ) ) )
17	10, 16	oveq12d 5871	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( A - B ) ) + ( B x. ( A - B ) ) ) = ( ( ( A ^ 2 ) - ( A x. B ) ) + ( ( A x. B ) - ( B ^ 2 ) ) ) )
18		sqcl 10537	. . . 4 |- ( A e. CC -> ( A ^ 2 ) e. CC )
19	18	adantr 274	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) e. CC )
20		mulcl 7901	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
21		sqcl 10537	. . . 4 |- ( B e. CC -> ( B ^ 2 ) e. CC )
22	21	adantl 275	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) e. CC )
23	19, 20, 22	npncand 8254	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) - ( A x. B ) ) + ( ( A x. B ) - ( B ^ 2 ) ) ) = ( ( A ^ 2 ) - ( B ^ 2 ) ) )
24	4, 17, 23	3eqtrrd 2208	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   + caddc 7777   x. cmul 7779   - cmin 8090   2c2 8929   ^cexp 10475

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476

This theorem is referenced by:  subsq2  10583  subsqi  10585  resqrexlemnm  10982  resqrexlemglsq  10986  pythagtriplem4  12222  pythagtriplem6  12224  pythagtriplem7  12225  pythagtriplem12  12229  pythagtriplem14  12231  pythagtriplem16  12233  difsqpwdvds  12291  4sqlem8  12337  4sqlem10  12339  lgslem1  13695  2sqlem4  13748