Theorem hisubcom 8891

Description: Two vector subtractions simultaneously commute in an inner product.

Hypotheses

Ref	Expression
hisubcom.1	|- A e. H~
hisubcom.2	|- B e. H~
hisubcom.3	|- C e. H~
hisubcom.4	|- D e. H~

Assertion

Ref	Expression
hisubcom	|- ((A -h B) .ih (C -h D)) = ((B -h A) .ih (D -h C))

Proof of Theorem hisubcom

Step	Hyp	Ref	Expression
1		hisubcom.2	. . . 4 |- B e. H~
2		hisubcom.1	. . . 4 |- A e. H~
3	1, 2	hvnegdi 8850	. . 3 |- (-u1 .h (B -h A)) = (A -h B)
4		hisubcom.4	. . . 4 |- D e. H~
5		hisubcom.3	. . . 4 |- C e. H~
6	4, 5	hvnegdi 8850	. . 3 |- (-u1 .h (D -h C)) = (C -h D)
7	3, 6	opreq12i 3958	. 2 |- ((-u1 .h (B -h A)) .ih (-u1 .h (D -h C))) = ((A -h B) .ih (C -h D))
8		ax1cn 5241	. . . . 5 |- 1 e. CC
9	8	negcl 5341	. . . 4 |- -u1 e. CC
10	1, 2	hvsubcl 8812	. . . 4 |- (B -h A) e. H~
11	4, 5	hvsubcl 8812	. . . 4 |- (D -h C) e. H~
12	9, 9, 10, 11	his35 8876	. . 3 |- ((-u1 .h (B -h A)) .ih (-u1 .h (D -h C))) = ((-u1 x. (*` -u1)) x. ((B -h A) .ih (D -h C)))
13		1re 5407	. . . . . . . 8 |- 1 e. RR
14	13	renegcl 5388	. . . . . . 7 |- -u1 e. RR
15		cjret 6745	. . . . . . 7 |- (-u1 e. RR -> (*` -u1) = -u1)
16	14, 15	ax-mp 7	. . . . . 6 |- (*` -u1) = -u1
17	16	opreq2i 3957	. . . . 5 |- (-u1 x. (*` -u1)) = (-u1 x. -u1)
18	8, 8	mul2neg 5419	. . . . 5 |- (-u1 x. -u1) = (1 x. 1)
19	8	mulid1 5304	. . . . 5 |- (1 x. 1) = 1
20	17, 18, 19	3eqtr 1491	. . . 4 |- (-u1 x. (*` -u1)) = 1
21	20	opreq1i 3956	. . 3 |- ((-u1 x. (*` -u1)) x. ((B -h A) .ih (D -h C))) = (1 x. ((B -h A) .ih (D -h C)))
22	10, 11	hicl 8869	. . . 4 |- ((B -h A) .ih (D -h C)) e. CC
23	22	mulid2 5305	. . 3 |- (1 x. ((B -h A) .ih (D -h C))) = ((B -h A) .ih (D -h C))
24	12, 21, 23	3eqtr 1491	. 2 |- ((-u1 .h (B -h A)) .ih (-u1 .h (D -h C))) = ((B -h A) .ih (D -h C))
25	7, 24	eqtr3 1489	1 |- ((A -h B) .ih (C -h D)) = ((B -h A) .ih (D -h C))

Syntax hints:   = wceq 953   e. wcel 955  ` cfv 3172  (class class class)co 3948  RRcr 5205  1c1 5207   x. cmul 5211  -ucneg 5265  *ccj 6680  H~chil 8727   .h csm 8729   -h cmv 8731   .ih csp 8732

This theorem is referenced by:  lnophmlem2 9857

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hfvadd 8791  ax-hvcom 8792  ax-hfvmul 8796  ax-hvmulid 8797  ax-hvmulass 8798  ax-hvdistr1 8799  ax-hfi 8867  ax-his1 8870  ax-his3 8872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-re 6682  df-im 6683  df-cj 6684  df-hvsub 8779

Copyright terms: Public domain