normpar2i - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > normpar2i		Structured version Visualization version GIF version

Theorem normpar2i 31231

Description: Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
normpar2.1	⊢ 𝐴 ∈ ℋ
normpar2.2	⊢ 𝐵 ∈ ℋ
normpar2.3	⊢ 𝐶 ∈ ℋ

**Assertion**
Ref	Expression
normpar2i	⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) = (((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) − ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2))

**Proof of Theorem normpar2i**
Step	Hyp	Ref	Expression
1		normpar2.1	. . . . . . 7 ⊢ 𝐴 ∈ ℋ
2		normpar2.2	. . . . . . 7 ⊢ 𝐵 ∈ ℋ
3	1, 2	hvaddcli 31093	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
4		2cn 12220	. . . . . . 7 ⊢ 2 ∈ ℂ
5		normpar2.3	. . . . . . 7 ⊢ 𝐶 ∈ ℋ
6	4, 5	hvmulcli 31089	. . . . . 6 ⊢ (2 ·_ℎ 𝐶) ∈ ℋ
7	3, 6	hvsubcli 31096	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
8	7	normcli 31206	. . . 4 ⊢ (norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))) ∈ ℝ
9	8	resqcli 14109	. . 3 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℝ
10	9	recni 11146	. 2 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℂ
11	1, 2	hvsubcli 31096	. . . . 5 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
12	11	normcli 31206	. . . 4 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐵)) ∈ ℝ
13	12	resqcli 14109	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℝ
14	13	recni 11146	. 2 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℂ
15		4cn 12230	. . . . 5 ⊢ 4 ∈ ℂ
16	1, 5	hvsubcli 31096	. . . . . . . 8 ⊢ (𝐴 −_ℎ 𝐶) ∈ ℋ
17	16	normcli 31206	. . . . . . 7 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℝ
18	17	resqcli 14109	. . . . . 6 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℝ
19	18	recni 11146	. . . . 5 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℂ
20	15, 19	mulcli 11139	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) ∈ ℂ
21	2, 5	hvsubcli 31096	. . . . . . . 8 ⊢ (𝐵 −_ℎ 𝐶) ∈ ℋ
22	21	normcli 31206	. . . . . . 7 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℝ
23	22	resqcli 14109	. . . . . 6 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℝ
24	23	recni 11146	. . . . 5 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℂ
25	15, 24	mulcli 11139	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) ∈ ℂ
26		2ne0 12249	. . . 4 ⊢ 2 ≠ 0
27	20, 25, 4, 26	divdiri 11898	. . 3 ⊢ (((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) / 2) = (((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) + ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2))
28	20, 25	addcomi 11324	. . . . . . 7 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) = ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) + (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)))
29		neg1cn 12130	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℂ
30	29, 6	hvmulcli 31089	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
31	29, 11	hvmulcli 31089	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) ∈ ℋ
32	3, 30, 31	hvadd32i 31129	. . . . . . . . . . . . . 14 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
33	3, 6	hvsubvali 31095	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
34	33	oveq1i 7368	. . . . . . . . . . . . . 14 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)))
35	4, 2	hvmulcli 31089	. . . . . . . . . . . . . . . 16 ⊢ (2 ·_ℎ 𝐵) ∈ ℋ
36	35, 6	hvsubvali 31095	. . . . . . . . . . . . . . 15 ⊢ ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
37	1, 2	hvcomi 31094	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 +_ℎ 𝐵) = (𝐵 +_ℎ 𝐴)
38	1, 2	hvnegdii 31137	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) = (𝐵 −_ℎ 𝐴)
39	37, 38	oveq12i 7370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴))
40	2, 1	hvsubcan2i 31139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴)) = (2 ·_ℎ 𝐵)
41	39, 40	eqtri 2759	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = (2 ·_ℎ 𝐵)
42	41	oveq1i 7368	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) = ((2 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
43	36, 42	eqtr4i 2762	. . . . . . . . . . . . . 14 ⊢ ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
44	32, 34, 43	3eqtr4i 2769	. . . . . . . . . . . . 13 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))
45	7, 11	hvsubvali 31095	. . . . . . . . . . . . 13 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)) = (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)))
46	4, 2, 5	hvsubdistr1i 31127	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))
47	44, 45, 46	3eqtr4i 2769	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐵 −_ℎ 𝐶))
48	47	fveq2i 6837	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵))) = (norm_ℎ‘(2 ·_ℎ (𝐵 −_ℎ 𝐶)))
49	4, 21	norm-iii-i 31214	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐵 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
50		0le2 12247	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
51		2re 12219	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
52	51	absidi 15301	. . . . . . . . . . . . 13 ⊢ (0 ≤ 2 → (abs‘2) = 2)
53	50, 52	ax-mp 5	. . . . . . . . . . . 12 ⊢ (abs‘2) = 2
54	53	oveq1i 7368	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
55	48, 49, 54	3eqtri 2763	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
56	55	oveq1i 7368	. . . . . . . . 9 ⊢ ((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)))↑2) = ((2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))↑2)
57	22	recni 11146	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℂ
58	4, 57	sqmuli 14107	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
59		sq2 14120	. . . . . . . . . 10 ⊢ (2↑2) = 4
60	59	oveq1i 7368	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
61	56, 58, 60	3eqtri 2763	. . . . . . . 8 ⊢ ((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)))↑2) = (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
62	1, 2	hvsubcan2i 31139	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ 𝐴)
63	62	oveq1i 7368	. . . . . . . . . . . . . 14 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (𝐴 −_ℎ 𝐵)) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) = ((2 ·_ℎ 𝐴) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
64	3, 30, 11	hvadd32i 31129	. . . . . . . . . . . . . 14 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (𝐴 −_ℎ 𝐵)) = (((𝐴 +_ℎ 𝐵) +_ℎ (𝐴 −_ℎ 𝐵)) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
65	4, 1	hvmulcli 31089	. . . . . . . . . . . . . . 15 ⊢ (2 ·_ℎ 𝐴) ∈ ℋ
66	65, 6	hvsubvali 31095	. . . . . . . . . . . . . 14 ⊢ ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
67	63, 64, 66	3eqtr4i 2769	. . . . . . . . . . . . 13 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (𝐴 −_ℎ 𝐵)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
68	33	oveq1i 7368	. . . . . . . . . . . . 13 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)) = (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (𝐴 −_ℎ 𝐵))
69	4, 1, 5	hvsubdistr1i 31127	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐴 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
70	67, 68, 69	3eqtr4i 2769	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐴 −_ℎ 𝐶))
71	70	fveq2i 6837	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵))) = (norm_ℎ‘(2 ·_ℎ (𝐴 −_ℎ 𝐶)))
72	4, 16	norm-iii-i 31214	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐴 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
73	53	oveq1i 7368	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
74	71, 72, 73	3eqtri 2763	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
75	74	oveq1i 7368	. . . . . . . . 9 ⊢ ((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)))↑2) = ((2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))↑2)
76	17	recni 11146	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℂ
77	4, 76	sqmuli 14107	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
78	59	oveq1i 7368	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
79	75, 77, 78	3eqtri 2763	. . . . . . . 8 ⊢ ((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)))↑2) = (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
80	61, 79	oveq12i 7370	. . . . . . 7 ⊢ (((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)))↑2) + ((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)))↑2)) = ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) + (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)))
81	28, 80	eqtr4i 2762	. . . . . 6 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) = (((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)))↑2) + ((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)))↑2))
82	7, 11	normpari 31229	. . . . . 6 ⊢ (((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)))↑2) + ((norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)))↑2)) = ((2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) + (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)))
83	81, 82	eqtri 2759	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) = ((2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) + (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)))
84	83	oveq1i 7368	. . . 4 ⊢ (((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) / 2) = (((2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) + (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2))) / 2)
85	4, 10	mulcli 11139	. . . . 5 ⊢ (2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) ∈ ℂ
86	4, 14	mulcli 11139	. . . . 5 ⊢ (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) ∈ ℂ
87	85, 86, 4, 26	divdiri 11898	. . . 4 ⊢ (((2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) + (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2))) / 2) = (((2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) / 2) + ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) / 2))
88	10, 4, 26	divcan3i 11887	. . . . 5 ⊢ ((2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) / 2) = ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)
89	14, 4, 26	divcan3i 11887	. . . . 5 ⊢ ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) / 2) = ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)
90	88, 89	oveq12i 7370	. . . 4 ⊢ (((2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) / 2) + ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) / 2)) = (((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) + ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2))
91	84, 87, 90	3eqtri 2763	. . 3 ⊢ (((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) / 2) = (((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) + ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2))
92	15, 19, 4, 26	div23i 11899	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
93		4div2e2 12310	. . . . . 6 ⊢ (4 / 2) = 2
94	93	oveq1i 7368	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
95	92, 94	eqtri 2759	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
96	15, 24, 4, 26	div23i 11899	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
97	93	oveq1i 7368	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
98	96, 97	eqtri 2759	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
99	95, 98	oveq12i 7370	. . 3 ⊢ (((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) + ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2)) = ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)))
100	27, 91, 99	3eqtr3i 2767	. 2 ⊢ (((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) + ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) = ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)))
101	10, 14, 100	mvlladdi 11399	1 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) = (((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) + (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))) − ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2))

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 ≤ cle 11167 − cmin 11364 -cneg 11365 / cdiv 11794 2c2 12200 4c4 12202 ↑cexp 13984 abscabs 15157 ℋchba 30994 +_ℎ cva 30995 ·_ℎ csm 30996 norm_ℎcno 30998 −_ℎ cmv 31000

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-hfvadd 31075 ax-hvcom 31076 ax-hvass 31077 ax-hv0cl 31078 ax-hvaddid 31079 ax-hfvmul 31080 ax-hvmulid 31081 ax-hvmulass 31082 ax-hvdistr1 31083 ax-hvdistr2 31084 ax-hvmul0 31085 ax-hfi 31154 ax-his1 31157 ax-his2 31158 ax-his3 31159 ax-his4 31160

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-n0 12402 df-z 12489 df-uz 12752 df-rp 12906 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-hnorm 31043 df-hvsub 31046

This theorem is referenced by: (None)

W3C validator