Theorem mulsub 6644

Description: Product of two differences.

Assertion

Ref	Expression
mulsub	|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A - B) x. (C - D)) = (((A x. C) + (D x. B)) - ((A x. D) + (C x. B))))

Proof of Theorem mulsub

Step	Hyp	Ref	Expression
1		negsub 6540	. . 3 |- ((A e. CC /\ B e. CC) -> (A + -uB) = (A - B))
2		negsub 6540	. . 3 |- ((C e. CC /\ D e. CC) -> (C + -uD) = (C - D))
3	1, 2	opreqan12d 4902	. 2 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + -uB) x. (C + -uD)) = ((A - B) x. (C - D)))
4		muladd 6582	. . . . 5 |- (((A e. CC /\ -uB e. CC) /\ (C e. CC /\ -uD e. CC)) -> ((A + -uB) x. (C + -uD)) = (((A x. C) + (-uD x. -uB)) + ((A x. -uD) + (C x. -uB))))
5		negcl 6525	. . . . 5 |- (D e. CC -> -uD e. CC)
6	4, 5	sylanr2 512	. . . 4 |- (((A e. CC /\ -uB e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + -uB) x. (C + -uD)) = (((A x. C) + (-uD x. -uB)) + ((A x. -uD) + (C x. -uB))))
7		negcl 6525	. . . 4 |- (B e. CC -> -uB e. CC)
8	6, 7	sylanl2 510	. . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + -uB) x. (C + -uD)) = (((A x. C) + (-uD x. -uB)) + ((A x. -uD) + (C x. -uB))))
9		mul2neg 6618	. . . . . . 7 |- ((D e. CC /\ B e. CC) -> (-uD x. -uB) = (D x. B))
10	9	ancoms 484	. . . . . 6 |- ((B e. CC /\ D e. CC) -> (-uD x. -uB) = (D x. B))
11	10	opreq2d 4898	. . . . 5 |- ((B e. CC /\ D e. CC) -> ((A x. C) + (-uD x. -uB)) = ((A x. C) + (D x. B)))
12	11	ad2ant2l 444	. . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. C) + (-uD x. -uB)) = ((A x. C) + (D x. B)))
13		mulneg2 6616	. . . . . . . 8 |- ((A e. CC /\ D e. CC) -> (A x. -uD) = -u(A x. D))
14		mulneg2 6616	. . . . . . . 8 |- ((C e. CC /\ B e. CC) -> (C x. -uB) = -u(C x. B))
15	13, 14	opreqan12d 4902	. . . . . . 7 |- (((A e. CC /\ D e. CC) /\ (C e. CC /\ B e. CC)) -> ((A x. -uD) + (C x. -uB)) = (-u(A x. D) + -u(C x. B)))
16		negdi 6620	. . . . . . . 8 |- (((A x. D) e. CC /\ (C x. B) e. CC) -> -u((A x. D) + (C x. B)) = (-u(A x. D) + -u(C x. B)))
17		mulcl 6456	. . . . . . . 8 |- ((A e. CC /\ D e. CC) -> (A x. D) e. CC)
18		mulcl 6456	. . . . . . . 8 |- ((C e. CC /\ B e. CC) -> (C x. B) e. CC)
19	16, 17, 18	syl2an 503	. . . . . . 7 |- (((A e. CC /\ D e. CC) /\ (C e. CC /\ B e. CC)) -> -u((A x. D) + (C x. B)) = (-u(A x. D) + -u(C x. B)))
20	15, 19	eqtr4d 1928	. . . . . 6 |- (((A e. CC /\ D e. CC) /\ (C e. CC /\ B e. CC)) -> ((A x. -uD) + (C x. -uB)) = -u((A x. D) + (C x. B)))
21	20	ancom2s 545	. . . . 5 |- (((A e. CC /\ D e. CC) /\ (B e. CC /\ C e. CC)) -> ((A x. -uD) + (C x. -uB)) = -u((A x. D) + (C x. B)))
22	21	an42s 567	. . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. -uD) + (C x. -uB)) = -u((A x. D) + (C x. B)))
23	12, 22	opreq12d 4900	. . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A x. C) + (-uD x. -uB)) + ((A x. -uD) + (C x. -uB))) = (((A x. C) + (D x. B)) + -u((A x. D) + (C x. B))))
24		addcl 6454	. . . . . 6 |- (((A x. C) e. CC /\ (D x. B) e. CC) -> ((A x. C) + (D x. B)) e. CC)
25		mulcl 6456	. . . . . 6 |- ((A e. CC /\ C e. CC) -> (A x. C) e. CC)
26		mulcl 6456	. . . . . . 7 |- ((D e. CC /\ B e. CC) -> (D x. B) e. CC)
27	26	ancoms 484	. . . . . 6 |- ((B e. CC /\ D e. CC) -> (D x. B) e. CC)
28	24, 25, 27	syl2an 503	. . . . 5 |- (((A e. CC /\ C e. CC) /\ (B e. CC /\ D e. CC)) -> ((A x. C) + (D x. B)) e. CC)
29	28	an4s 566	. . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. C) + (D x. B)) e. CC)
30		addcl 6454	. . . . . 6 |- (((A x. D) e. CC /\ (C x. B) e. CC) -> ((A x. D) + (C x. B)) e. CC)
31	18	ancoms 484	. . . . . 6 |- ((B e. CC /\ C e. CC) -> (C x. B) e. CC)
32	30, 17, 31	syl2an 503	. . . . 5 |- (((A e. CC /\ D e. CC) /\ (B e. CC /\ C e. CC)) -> ((A x. D) + (C x. B)) e. CC)
33	32	an42s 567	. . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. D) + (C x. B)) e. CC)
34		negsub 6540	. . . 4 |- ((((A x. C) + (D x. B)) e. CC /\ ((A x. D) + (C x. B)) e. CC) -> (((A x. C) + (D x. B)) + -u((A x. D) + (C x. B))) = (((A x. C) + (D x. B)) - ((A x. D) + (C x. B))))
35	29, 33, 34	syl11anc 524	. . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A x. C) + (D x. B)) + -u((A x. D) + (C x. B))) = (((A x. C) + (D x. B)) - ((A x. D) + (C x. B))))
36	8, 23, 35	3eqtrd 1929	. 2 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + -uB) x. (C + -uD)) = (((A x. C) + (D x. B)) - ((A x. D) + (C x. B))))
37	3, 36	eqtr3d 1927	1 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A - B) x. (C - D)) = (((A x. C) + (D x. B)) - ((A x. D) + (C x. B))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  (class class class)co 4884  CCcc 6384   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446

This theorem is referenced by:  muleqadd 6889  addltmul 7229  sqabssub 8100  sinaddi 8716  cosaddi 8717  addltmulALT 12018

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-sub 6511  df-neg 6513

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