Theorem normpar2i 10656

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
normpar2i	|- ((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 normpar2i

Step	Hyp	Ref	Expression
1		2re 7163	. . . . . 6 |- 2 e. RR
2		normpar2.1	. . . . . . . . 9 |- A e. ~H
3		normpar2.3	. . . . . . . . 9 |- C e. ~H
4	2, 3	hvsubcli 10523	. . . . . . . 8 |- (A -h C) e. ~H
5	4	normcli 10631	. . . . . . 7 |- (normh` (A -h C)) e. RR
6	5	resqcli 7868	. . . . . 6 |- ((normh` (A -h C))^2) e. RR
7	1, 6	remulcli 6488	. . . . 5 |- (2 x. ((normh` (A -h C))^2)) e. RR
8		normpar2.2	. . . . . . . . 9 |- B e. ~H
9	8, 3	hvsubcli 10523	. . . . . . . 8 |- (B -h C) e. ~H
10	9	normcli 10631	. . . . . . 7 |- (normh` (B -h C)) e. RR
11	10	resqcli 7868	. . . . . 6 |- ((normh` (B -h C))^2) e. RR
12	1, 11	remulcli 6488	. . . . 5 |- (2 x. ((normh` (B -h C))^2)) e. RR
13	7, 12	readdcli 6487	. . . 4 |- ((2 x. ((normh` (A -h C))^2)) + (2 x. ((normh` (B -h C))^2))) e. RR
14	13	recni 6467	. . 3 |- ((2 x. ((normh` (A -h C))^2)) + (2 x. ((normh` (B -h C))^2))) e. CC
15	2, 8	hvaddcli 10520	. . . . . . 7 |- (A +h B) e. ~H
16		2cn 7164	. . . . . . . 8 |- 2 e. CC
17	16, 3	hvmulcli 10516	. . . . . . 7 |- (2 .h C) e. ~H
18	15, 17	hvsubcli 10523	. . . . . 6 |- ((A +h B) -h (2 .h C)) e. ~H
19	18	normcli 10631	. . . . 5 |- (normh` ((A +h B) -h (2 .h C))) e. RR
20	19	resqcli 7868	. . . 4 |- ((normh` ((A +h B) -h (2 .h C)))^2) e. RR
21	20	recni 6467	. . 3 |- ((normh` ((A +h B) -h (2 .h C)))^2) e. CC
22	2, 8	hvsubcli 10523	. . . . . 6 |- (A -h B) e. ~H
23	22	normcli 10631	. . . . 5 |- (normh` (A -h B)) e. RR
24	23	resqcli 7868	. . . 4 |- ((normh` (A -h B))^2) e. RR
25	24	recni 6467	. . 3 |- ((normh` (A -h B))^2) e. CC
26		4re 7166	. . . . . . 7 |- 4 e. RR
27	26	recni 6467	. . . . . 6 |- 4 e. CC
28	6	recni 6467	. . . . . 6 |- ((normh` (A -h C))^2) e. CC
29	27, 28	mulcli 6474	. . . . 5 |- (4 x. ((normh` (A -h C))^2)) e. CC
30	11	recni 6467	. . . . . 6 |- ((normh` (B -h C))^2) e. CC
31	27, 30	mulcli 6474	. . . . 5 |- (4 x. ((normh` (B -h C))^2)) e. CC
32		2ne0 7174	. . . . 5 |- 2 =/= 0
33	29, 31, 16, 32	divdiri 6930	. . . 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	addcomi 6475	. . . . . . 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	hvsubvali 10522	. . . . . . . . . . . . . 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	hvsubvali 10522	. . . . . . . . . . . . . . 15 |- ((A +h B) -h (2 .h C)) = ((A +h B) +h (-u1 .h (2 .h C)))
37	36	opreq1i 4892	. . . . . . . . . . . . . 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	hvcomi 10521	. . . . . . . . . . . . . . . . . 18 |- (A +h B) = (B +h A)
39	2, 8	hvnegdii 10561	. . . . . . . . . . . . . . . . . 18 |- (-u1 .h (A -h B)) = (B -h A)
40	38, 39	opreq12i 4894	. . . . . . . . . . . . . . . . 17 |- ((A +h B) +h (-u1 .h (A -h B))) = ((B +h A) +h (B -h A))
41	8, 2	hvsubcan2i 10563	. . . . . . . . . . . . . . . . 17 |- ((B +h A) +h (B -h A)) = (2 .h B)
42	40, 41	eqtri 1908	. . . . . . . . . . . . . . . 16 |- ((A +h B) +h (-u1 .h (A -h B))) = (2 .h B)
43	42	opreq1i 4892	. . . . . . . . . . . . . . 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 6422	. . . . . . . . . . . . . . . . . 18 |- 1 e. CC
45	44	negcli 6526	. . . . . . . . . . . . . . . . 17 |- -u1 e. CC
46	45, 17	hvmulcli 10516	. . . . . . . . . . . . . . . 16 |- (-u1 .h (2 .h C)) e. ~H
47	45, 22	hvmulcli 10516	. . . . . . . . . . . . . . . 16 |- (-u1 .h (A -h B)) e. ~H
48	15, 46, 47	hvadd23i 10553	. . . . . . . . . . . . . . 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	hvmulcli 10516	. . . . . . . . . . . . . . . 16 |- (2 .h B) e. ~H
50	49, 17	hvsubvali 10522	. . . . . . . . . . . . . . 15 |- ((2 .h B) -h (2 .h C)) = ((2 .h B) +h (-u1 .h (2 .h C)))
51	43, 48, 50	3eqtr4i 1921	. . . . . . . . . . . . . 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	3eqtri 1912	. . . . . . . . . . . . 13 |- (((A +h B) -h (2 .h C)) -h (A -h B)) = ((2 .h B) -h (2 .h C))
53	16, 8, 3	hvsubdistr1i 10551	. . . . . . . . . . . . 13 |- (2 .h (B -h C)) = ((2 .h B) -h (2 .h C))
54	52, 53	eqtr4i 1911	. . . . . . . . . . . 12 |- (((A +h B) -h (2 .h C)) -h (A -h B)) = (2 .h (B -h C))
55	54	fveq2i 4684	. . . . . . . . . . 11 |- (normh` (((A +h B) -h (2 .h C)) -h (A -h B))) = (normh` (2 .h (B -h C)))
56	16, 9	norm-iii.i 10639	. . . . . . . . . . 11 |- (normh` (2 .h (B -h C))) = ((abs` 2) x. (normh` (B -h C)))
57		0re 6603	. . . . . . . . . . . . . 14 |- 0 e. RR
58		2pos 7173	. . . . . . . . . . . . . 14 |- 0 < 2
59	57, 1, 58	ltleii 6756	. . . . . . . . . . . . 13 |- 0 <_ 2
60	1	absidi 8112	. . . . . . . . . . . . 13 |- (0 <_ 2 -> (abs` 2) = 2)
61	59, 60	ax-mp 7	. . . . . . . . . . . 12 |- (abs` 2) = 2
62	61	opreq1i 4892	. . . . . . . . . . 11 |- ((abs` 2) x. (normh` (B -h C))) = (2 x. (normh` (B -h C)))
63	55, 56, 62	3eqtri 1912	. . . . . . . . . 10 |- (normh` (((A +h B) -h (2 .h C)) -h (A -h B))) = (2 x. (normh` (B -h C)))
64	63	opreq1i 4892	. . . . . . . . 9 |- ((normh` (((A +h B) -h (2 .h C)) -h (A -h B)))^2) = ((2 x. (normh` (B -h C)))^2)
65	10	recni 6467	. . . . . . . . . 10 |- (normh` (B -h C)) e. CC
66	16, 65	sqmuli 7862	. . . . . . . . 9 |- ((2 x. (normh` (B -h C)))^2) = ((2^2) x. ((normh` (B -h C))^2))
67		sq2 7883	. . . . . . . . . 10 |- (2^2) = 4
68	67	opreq1i 4892	. . . . . . . . 9 |- ((2^2) x. ((normh` (B -h C))^2)) = (4 x. ((normh` (B -h C))^2))
69	64, 66, 68	3eqtri 1912	. . . . . . . 8 |- ((normh` (((A +h B) -h (2 .h C)) -h (A -h B)))^2) = (4 x. ((normh` (B -h C))^2))
70	36	opreq1i 4892	. . . . . . . . . . . . . 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	hvadd23i 10553	. . . . . . . . . . . . . 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	hvsubcan2i 10563	. . . . . . . . . . . . . . . 16 |- ((A +h B) +h (A -h B)) = (2 .h A)
73	72	opreq1i 4892	. . . . . . . . . . . . . . 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	hvmulcli 10516	. . . . . . . . . . . . . . . 16 |- (2 .h A) e. ~H
75	74, 17	hvsubvali 10522	. . . . . . . . . . . . . . 15 |- ((2 .h A) -h (2 .h C)) = ((2 .h A) +h (-u1 .h (2 .h C)))
76	73, 75	eqtr4i 1911	. . . . . . . . . . . . . 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	3eqtri 1912	. . . . . . . . . . . . 13 |- (((A +h B) -h (2 .h C)) +h (A -h B)) = ((2 .h A) -h (2 .h C))
78	16, 2, 3	hvsubdistr1i 10551	. . . . . . . . . . . . 13 |- (2 .h (A -h C)) = ((2 .h A) -h (2 .h C))
79	77, 78	eqtr4i 1911	. . . . . . . . . . . 12 |- (((A +h B) -h (2 .h C)) +h (A -h B)) = (2 .h (A -h C))
80	79	fveq2i 4684	. . . . . . . . . . 11 |- (normh` (((A +h B) -h (2 .h C)) +h (A -h B))) = (normh` (2 .h (A -h C)))
81	16, 4	norm-iii.i 10639	. . . . . . . . . . 11 |- (normh` (2 .h (A -h C))) = ((abs` 2) x. (normh` (A -h C)))
82	61	opreq1i 4892	. . . . . . . . . . 11 |- ((abs` 2) x. (normh` (A -h C))) = (2 x. (normh` (A -h C)))
83	80, 81, 82	3eqtri 1912	. . . . . . . . . 10 |- (normh` (((A +h B) -h (2 .h C)) +h (A -h B))) = (2 x. (normh` (A -h C)))
84	83	opreq1i 4892	. . . . . . . . 9 |- ((normh` (((A +h B) -h (2 .h C)) +h (A -h B)))^2) = ((2 x. (normh` (A -h C)))^2)
85	5	recni 6467	. . . . . . . . . 10 |- (normh` (A -h C)) e. CC
86	16, 85	sqmuli 7862	. . . . . . . . 9 |- ((2 x. (normh` (A -h C)))^2) = ((2^2) x. ((normh` (A -h C))^2))
87	67	opreq1i 4892	. . . . . . . . 9 |- ((2^2) x. ((normh` (A -h C))^2)) = (4 x. ((normh` (A -h C))^2))
88	84, 86, 87	3eqtri 1912	. . . . . . . 8 |- ((normh` (((A +h B) -h (2 .h C)) +h (A -h B)))^2) = (4 x. ((normh` (A -h C))^2))
89	69, 88	opreq12i 4894	. . . . . . 7 |- (((normh` (((A +h B) -h (2 .h C)) -h (A -h B)))^2) + ((normh` (((A +h B) -h (2 .h C)) +h (A -h B)))^2)) = ((4 x. ((normh` (B -h C))^2)) + (4 x. ((normh` (A -h C))^2)))
90	18, 22	normpari 10654	. . . . . . 7 |- (((normh` (((A +h B) -h (2 .h C)) -h (A -h B)))^2) + ((normh` (((A +h B) -h (2 .h C)) +h (A -h B)))^2)) = ((2 x. ((normh` ((A +h B) -h (2 .h C)))^2)) + (2 x. ((normh` (A -h B))^2)))
91	34, 89, 90	3eqtr2i 1915	. . . . . 6 |- ((4 x. ((normh` (A -h C))^2)) + (4 x. ((normh` (B -h C))^2))) = ((2 x. ((normh` ((A +h B) -h (2 .h C)))^2)) + (2 x. ((normh` (A -h B))^2)))
92	91	opreq1i 4892	. . . . 5 |- (((4 x. ((normh` (A -h C))^2)) + (4 x. ((normh` (B -h C))^2))) / 2) = (((2 x. ((normh` ((A +h B) -h (2 .h C)))^2)) + (2 x. ((normh` (A -h B))^2))) / 2)
93	16, 21	mulcli 6474	. . . . . 6 |- (2 x. ((normh` ((A +h B) -h (2 .h C)))^2)) e. CC
94	16, 25	mulcli 6474	. . . . . 6 |- (2 x. ((normh` (A -h B))^2)) e. CC
95	93, 94, 16, 32	divdiri 6930	. . . . 5 |- (((2 x. ((normh` ((A +h B) -h (2 .h C)))^2)) + (2 x. ((normh` (A -h B))^2))) / 2) = (((2 x. ((normh` ((A +h B) -h (2 .h C)))^2)) / 2) + ((2 x. ((normh` (A -h B))^2)) / 2))
96	21, 16, 32	divcan3i 6934	. . . . . 6 |- ((2 x. ((normh` ((A +h B) -h (2 .h C)))^2)) / 2) = ((normh` ((A +h B) -h (2 .h C)))^2)
97	25, 16, 32	divcan3i 6934	. . . . . 6 |- ((2 x. ((normh` (A -h B))^2)) / 2) = ((normh` (A -h B))^2)
98	96, 97	opreq12i 4894	. . . . 5 |- (((2 x. ((normh` ((A +h B) -h (2 .h C)))^2)) / 2) + ((2 x. ((normh` (A -h B))^2)) / 2)) = (((normh` ((A +h B) -h (2 .h C)))^2) + ((normh` (A -h B))^2))
99	92, 95, 98	3eqtri 1912	. . . 4 |- (((4 x. ((normh` (A -h C))^2)) + (4 x. ((normh` (B -h C))^2))) / 2) = (((normh` ((A +h B) -h (2 .h C)))^2) + ((normh` (A -h B))^2))
100	27, 28, 16, 32	div23i 6931	. . . . . 6 |- ((4 x. ((normh` (A -h C))^2)) / 2) = ((4 / 2) x. ((normh` (A -h C))^2))
101		4d2e2 7211	. . . . . . 7 |- (4 / 2) = 2
102	101	opreq1i 4892	. . . . . 6 |- ((4 / 2) x. ((normh` (A -h C))^2)) = (2 x. ((normh` (A -h C))^2))
103	100, 102	eqtri 1908	. . . . 5 |- ((4 x. ((normh` (A -h C))^2)) / 2) = (2 x. ((normh` (A -h C))^2))
104	27, 30, 16, 32	div23i 6931	. . . . . 6 |- ((4 x. ((normh` (B -h C))^2)) / 2) = ((4 / 2) x. ((normh` (B -h C))^2))
105	101	opreq1i 4892	. . . . . 6 |- ((4 / 2) x. ((normh` (B -h C))^2)) = (2 x. ((normh` (B -h C))^2))
106	104, 105	eqtri 1908	. . . . 5 |- ((4 x. ((normh` (B -h C))^2)) / 2) = (2 x. ((normh` (B -h C))^2))
107	103, 106	opreq12i 4894	. . . 4 |- (((4 x. ((normh` (A -h C))^2)) / 2) + ((4 x. ((normh` (B -h C))^2)) / 2)) = ((2 x. ((normh` (A -h C))^2)) + (2 x. ((normh` (B -h C))^2)))
108	33, 99, 107	3eqtr3i 1918	. . 3 |- (((normh` ((A +h B) -h (2 .h C)))^2) + ((normh` (A -h B))^2)) = ((2 x. ((normh` (A -h C))^2)) + (2 x. ((normh` (B -h C))^2)))
109	14, 21, 25, 108	subaddrii 6529	. 2 |- (((2 x. ((normh` (A -h C))^2)) + (2 x. ((normh` (B -h C))^2))) - ((normh` ((A +h B) -h (2 .h C)))^2)) = ((normh` (A -h B))^2)
110	109	eqcomi 1888	1 |- ((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))

Syntax hints:   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447   <_ cle 6448  2c2 7145  4c4 7147  ^cexp 7811  abscabs 8000  ~Hchil 10420   +h cva 10421   .h csm 10422   -h cmv 10424  normhcno 10426

This theorem is referenced by:  projlem5 10823

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  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585

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-nel 2020  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-sup 5664  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-ltr 6322  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-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-hnorm 10469  df-hvsub 10472

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