Theorem normpar2 9023

Description: Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100.

Hypotheses

Ref	Expression
normpar2.1	|- A e. H~
normpar2.2	|- B e. H~
normpar2.3	|- C e. H~

Assertion

Ref	Expression
normpar2	|- ((normh` (A -h B))^2) = (((2 x. ((normh` (A -h C))^2)) + (2 x. ((normh` (B -h C))^2))) - ((normh` ((A +h B) -h (2 .h C)))^2))

Proof of Theorem normpar2

Step	Hyp	Ref	Expression
1		2re 5979	. . . . . 6 |- 2 e. RR
2		normpar2.1	. . . . . . . . 9 |- A e. H~
3		normpar2.3	. . . . . . . . 9 |- C e. H~
4	2, 3	hvsubcl 8891	. . . . . . . 8 |- (A -h C) e. H~
5	4	normcl 8998	. . . . . . 7 |- (normh` (A -h C)) e. RR
6	5	resqcl 6623	. . . . . 6 |- ((normh` (A -h C))^2) e. RR
7	1, 6	remulcl 5335	. . . . 5 |- (2 x. ((normh` (A -h C))^2)) e. RR
8		normpar2.2	. . . . . . . . 9 |- B e. H~
9	8, 3	hvsubcl 8891	. . . . . . . 8 |- (B -h C) e. H~
10	9	normcl 8998	. . . . . . 7 |- (normh` (B -h C)) e. RR
11	10	resqcl 6623	. . . . . 6 |- ((normh` (B -h C))^2) e. RR
12	1, 11	remulcl 5335	. . . . 5 |- (2 x. ((normh` (B -h C))^2)) e. RR
13	7, 12	readdcl 5334	. . . 4 |- ((2 x. ((normh` (A -h C))^2)) + (2 x. ((normh` (B -h C))^2))) e. RR
14	13	recn 5314	. . 3 |- ((2 x. ((normh` (A -h C))^2)) + (2 x. ((normh` (B -h C))^2))) e. CC
15	2, 8	hvaddcl 8888	. . . . . . 7 |- (A +h B) e. H~
16		2cn 5980	. . . . . . . 8 |- 2 e. CC
17	16, 3	hvmulcl 8884	. . . . . . 7 |- (2 .h C) e. H~
18	15, 17	hvsubcl 8891	. . . . . 6 |- ((A +h B) -h (2 .h C)) e. H~
19	18	normcl 8998	. . . . 5 |- (normh` ((A +h B) -h (2 .h C))) e. RR
20	19	resqcl 6623	. . . 4 |- ((normh` ((A +h B) -h (2 .h C)))^2) e. RR
21	20	recn 5314	. . 3 |- ((normh` ((A +h B) -h (2 .h C)))^2) e. CC
22	2, 8	hvsubcl 8891	. . . . . 6 |- (A -h B) e. H~
23	22	normcl 8998	. . . . 5 |- (normh` (A -h B)) e. RR
24	23	resqcl 6623	. . . 4 |- ((normh` (A -h B))^2) e. RR
25	24	recn 5314	. . 3 |- ((normh` (A -h B))^2) e. CC
26		4re 5982	. . . . . . 7 |- 4 e. RR
27	26	recn 5314	. . . . . 6 |- 4 e. CC
28	6	recn 5314	. . . . . 6 |- ((normh` (A -h C))^2) e. CC
29	27, 28	mulcl 5321	. . . . 5 |- (4 x. ((normh` (A -h C))^2)) e. CC
30	11	recn 5314	. . . . . 6 |- ((normh` (B -h C))^2) e. CC
31	27, 30	mulcl 5321	. . . . 5 |- (4 x. ((normh` (B -h C))^2)) e. CC
32		2ne0 5990	. . . . 5 |- 2 =/= 0
33	29, 31, 16, 32	divdir 5747	. . . 4 |- (((4 x. ((normh` (A -h C))^2)) + (4 x. ((normh` (B -h C))^2))) / 2) = (((4 x. ((normh` (A -h C))^2)) / 2) + ((4 x. ((normh` (B -h C))^2)) / 2))
34	29, 31	addcom 5322	. . . . . . 7 |- ((4 x. ((normh` (A -h C))^2)) + (4 x. ((normh` (B -h C))^2))) = ((4 x. ((normh` (B -h C))^2)) + (4 x. ((normh` (A -h C))^2)))
35	18, 22	hvsubval 8890	. . . . . . . . . . . . . 14 |- (((A +h B) -h (2 .h C)) -h (A -h B)) = (((A +h B) -h (2 .h C)) +h (-u1 .h (A -h B)))
36	15, 17	hvsubval 8890	. . . . . . . . . . . . . . 15 |- ((A +h B) -h (2 .h C)) = ((A +h B) +h (-u1 .h (2 .h C)))
37	36	opreq1i 3971	. . . . . . . . . . . . . 14 |- (((A +h B) -h (2 .h C)) +h (-u1 .h (A -h B))) = (((A +h B) +h (-u1 .h (2 .h C))) +h (-u1 .h (A -h B)))
38	2, 8	hvcom 8889	. . . . . . . . . . . . . . . . . 18 |- (A +h B) = (B +h A)
39	2, 8	hvnegdi 8929	. . . . . . . . . . . . . . . . . 18 |- (-u1 .h (A -h B)) = (B -h A)
40	38, 39	opreq12i 3973	. . . . . . . . . . . . . . . . 17 |- ((A +h B) +h (-u1 .h (A -h B))) = ((B +h A) +h (B -h A))
41	8, 2	hvsubcan2 8931	. . . . . . . . . . . . . . . . 17 |- ((B +h A) +h (B -h A)) = (2 .h B)
42	40, 41	eqtr 1495	. . . . . . . . . . . . . . . 16 |- ((A +h B) +h (-u1 .h (A -h B))) = (2 .h B)
43	42	opreq1i 3971	. . . . . . . . . . . . . . 15 |- (((A +h B) +h (-u1 .h (A -h B))) +h (-u1 .h (2 .h C))) = ((2 .h B) +h (-u1 .h (2 .h C)))
44		ax1cn 5269	. . . . . . . . . . . . . . . . . 18 |- 1 e. CC
45	44	negcl 5369	. . . . . . . . . . . . . . . . 17 |- -u1 e. CC
46	45, 17	hvmulcl 8884	. . . . . . . . . . . . . . . 16 |- (-u1 .h (2 .h C)) e. H~
47	45, 22	hvmulcl 8884	. . . . . . . . . . . . . . . 16 |- (-u1 .h (A -h B)) e. H~
48	15, 46, 47	hvadd23 8921	. . . . . . . . . . . . . . 15 |- (((A +h B) +h (-u1 .h (2 .h C))) +h (-u1 .h (A -h B))) = (((A +h B) +h (-u1 .h (A -h B))) +h (-u1 .h (2 .h C)))
49	16, 8	hvmulcl 8884	. . . . . . . . . . . . . . . 16 |- (2 .h B) e. H~
50	49, 17	hvsubval 8890	. . . . . . . . . . . . . . 15 |- ((2 .h B) -h (2 .h C)) = ((2 .h B) +h (-u1 .h (2 .h C)))
51	43, 48, 50	3eqtr4 1505	. . . . . . . . . . . . . 14 |- (((A +h B) +h (-u1 .h (2 .h C))) +h (-u1 .h (A -h B))) = ((2 .h B) -h (2 .h C))
52	35, 37, 51	3eqtr 1499	. . . . . . . . . . . . 13 |- (((A +h B) -h (2 .h C)) -h (A -h B)) = ((2 .h B) -h (2 .h C))
53	16, 8, 3	hvsubdistr1 8919	. . . . . . . . . . . . 13 |- (2 .h (B -h C)) = ((2 .h B) -h (2 .h C))
54	52, 53	eqtr4 1498	. . . . . . . . . . . 12 |- (((A +h B) -h (2 .h C)) -h (A -h B)) = (2 .h (B -h C))
55	54	fveq2i 3727	. . . . . . . . . . 11 |- (normh` (((A +h B) -h (2 .h C)) -h (A -h B))) = (normh` (2 .h (B -h C)))
56	16, 9	norm-iii 9006	. . . . . . . . . . 11 |- (normh` (2 .h (B -h C))) = ((abs` 2) x. (normh` (B -h C)))
57		0re 5440	. . . . . . . . . . . . . 14 |- 0 e. RR
58		2pos 5989	. . . . . . . . . . . . . 14 |- 0 < 2
59	57, 1, 58	ltlei 5581	. . . . . . . . . . . . 13 |- 0 <_ 2
60	1	absid 6861	. . . . . . . . . . . . 13 |- (0 <_ 2 -> (abs` 2) = 2)
61	59, 60	ax-mp 7	. . . . . . . . . . . 12 |- (abs` 2) = 2
62	61	opreq1i 3971	. . . . . . . . . . 11 |- ((abs` 2) x. (normh` (B -h C))) = (2 x. (normh` (B -h C)))
63	55, 56, 62	3eqtr 1499	. . . . . . . . . 10 |- (normh` (((A +h B) -h (2 .h C)) -h (A -h B))) = (2 x. (normh` (B -h C)))
64	63	opreq1i 3971	. . . . . . . . 9 |- ((normh` (((A +h B) -h (2 .h C)) -h (A -h B)))^2) = ((2 x. (normh` (B -h C)))^2)
65	10	recn 5314	. . . . . . . . . 10 |- (normh` (B -h C)) e. CC
66	16, 65	sqmul 6617	. . . . . . . . 9 |- ((2 x. (normh` (B -h C)))^2) = ((2^2) x. ((normh` (B -h C))^2))
67		sq2 6638	. . . . . . . . . 10 |- (2^2) = 4
68	67	opreq1i 3971	. . . . . . . . 9 |- ((2^2) x. ((normh` (B -h C))^2)) = (4 x. ((normh` (B -h C))^2))
69	64, 66, 68	3eqtr 1499	. . . . . . . 8 |- ((normh` (((A +h B) -h (2 .h C)) -h (A -h B)))^2) = (4 x. ((normh` (B -h C))^2))
70	36	opreq1i 3971	. . . . . . . . . . . . . 14 |- (((A +h B) -h (2 .h C)) +h (A -h B)) = (((A +h B) +h (-u1 .h (2 .h C))) +h (A -h B))
71	15, 46, 22	hvadd23 8921	. . . . . . . . . . . . . 14 |- (((A +h B) +h (-u1 .h (2 .h C))) +h (A -h B)) = (((A +h B) +h (A -h B)) +h (-u1 .h (2 .h C)))
72	2, 8	hvsubcan2 8931	. . . . . . . . . . . . . . . 16 |- ((A +h B) +h (A -h B)) = (2 .h A)
73	72	opreq1i 3971	. . . . . . . . . . . . . . 15 |- (((A +h B) +h (A -h B)) +h (-u1 .h (2 .h C))) = ((2 .h A) +h (-u1 .h (2 .h C)))
74	16, 2	hvmulcl 8884	. . . . . . . . . . . . . . . 16 |- (2 .h A) e. H~
75	74, 17	hvsubval 8890	. . . . . . . . . . . . . . 15 |- ((2 .h A) -h (2 .h C)) = ((2 .h A) +h (-u1 .h (2 .h C)))
76	73, 75	eqtr4 1498	. . . . . . . . . . . . . 14 |- (((A +h B) +h (A -h B)) +h (-u1 .h (2 .h C))) = ((2 .h A) -h (2 .h C))
77	70, 71, 76	3eqtr 1499	. . . . . . . . . . . . 13 |- (((A +h B) -h (2 .h C)) +h (A -h B)) = ((2 .h A) -h (2 .h C))
78	16, 2, 3	hvsubdistr1 8919	. . . . . . . . . . . . 13 |- (2 .h (A -h C)) = ((2 .h A)