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 31508

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 31370	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
4		2cn 12311	. . . . . . 7 ⊢ 2 ∈ ℂ
5		normpar2.3	. . . . . . 7 ⊢ 𝐶 ∈ ℋ
6	4, 5	hvmulcli 31366	. . . . . 6 ⊢ (2 ·_ℎ 𝐶) ∈ ℋ
7	3, 6	hvsubcli 31373	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
8	7	normcli 31483	. . . 4 ⊢ (norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))) ∈ ℝ
9	8	resqcli 14218	. . 3 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℝ
10	9	recni 11218	. 2 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℂ
11	1, 2	hvsubcli 31373	. . . . 5 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
12	11	normcli 31483	. . . 4 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐵)) ∈ ℝ
13	12	resqcli 14218	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℝ
14	13	recni 11218	. 2 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℂ
15		4cn 12321	. . . . 5 ⊢ 4 ∈ ℂ
16	1, 5	hvsubcli 31373	. . . . . . . 8 ⊢ (𝐴 −_ℎ 𝐶) ∈ ℋ
17	16	normcli 31483	. . . . . . 7 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℝ
18	17	resqcli 14218	. . . . . 6 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℝ
19	18	recni 11218	. . . . 5 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℂ
20	15, 19	mulcli 11211	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) ∈ ℂ
21	2, 5	hvsubcli 31373	. . . . . . . 8 ⊢ (𝐵 −_ℎ 𝐶) ∈ ℋ
22	21	normcli 31483	. . . . . . 7 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℝ
23	22	resqcli 14218	. . . . . 6 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℝ
24	23	recni 11218	. . . . 5 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℂ
25	15, 24	mulcli 11211	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) ∈ ℂ
26		2ne0 12342	. . . 4 ⊢ 2 ≠ 0
27	20, 25, 4, 26	divdiri 11967	. . 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 11396	. . . . . . 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 12198	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℂ
30	29, 6	hvmulcli 31366	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
31	29, 11	hvmulcli 31366	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) ∈ ℋ
32	3, 30, 31	hvadd32i 31406	. . . . . . . . . . . . . 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 31372	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
34	33	oveq1i 7420	. . . . . . . . . . . . . 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 31366	. . . . . . . . . . . . . . . 16 ⊢ (2 ·_ℎ 𝐵) ∈ ℋ
36	35, 6	hvsubvali 31372	. . . . . . . . . . . . . . 15 ⊢ ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
37	1, 2	hvcomi 31371	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 +_ℎ 𝐵) = (𝐵 +_ℎ 𝐴)
38	1, 2	hvnegdii 31414	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) = (𝐵 −_ℎ 𝐴)
39	37, 38	oveq12i 7422	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴))
40	2, 1	hvsubcan2i 31416	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴)) = (2 ·_ℎ 𝐵)
41	39, 40	eqtri 2786	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = (2 ·_ℎ 𝐵)
42	41	oveq1i 7420	. . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . 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 31372	. . . . . . . . . . . . 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 31404	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))
47	44, 45, 46	3eqtr4i 2796	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐵 −_ℎ 𝐶))
48	47	fveq2i 6884	. . . . . . . . . . 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 31491	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐵 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
50		0le2 12338	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
51		2re 12310	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
52	51	absidi 15425	. . . . . . . . . . . . 13 ⊢ (0 ≤ 2 → (abs‘2) = 2)
53	50, 52	ax-mp 5	. . . . . . . . . . . 12 ⊢ (abs‘2) = 2
54	53	oveq1i 7420	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
55	48, 49, 54	3eqtri 2790	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
56	55	oveq1i 7420	. . . . . . . . 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 11218	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℂ
58	4, 57	sqmuli 14216	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
59		sq2 14229	. . . . . . . . . 10 ⊢ (2↑2) = 4
60	59	oveq1i 7420	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
61	56, 58, 60	3eqtri 2790	. . . . . . . 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 31416	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ 𝐴)
63	62	oveq1i 7420	. . . . . . . . . . . . . 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 31406	. . . . . . . . . . . . . 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 31366	. . . . . . . . . . . . . . 15 ⊢ (2 ·_ℎ 𝐴) ∈ ℋ
66	65, 6	hvsubvali 31372	. . . . . . . . . . . . . 14 ⊢ ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
67	63, 64, 66	3eqtr4i 2796	. . . . . . . . . . . . 13 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (𝐴 −_ℎ 𝐵)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
68	33	oveq1i 7420	. . . . . . . . . . . . 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 31404	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐴 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
70	67, 68, 69	3eqtr4i 2796	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐴 −_ℎ 𝐶))
71	70	fveq2i 6884	. . . . . . . . . . 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 31491	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐴 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
73	53	oveq1i 7420	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
74	71, 72, 73	3eqtri 2790	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
75	74	oveq1i 7420	. . . . . . . . 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 11218	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℂ
77	4, 76	sqmuli 14216	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
78	59	oveq1i 7420	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
79	75, 77, 78	3eqtri 2790	. . . . . . . 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 7422	. . . . . . 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 2789	. . . . . 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 31506	. . . . . 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 2786	. . . . 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 7420	. . . 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 11211	. . . . 5 ⊢ (2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) ∈ ℂ
86	4, 14	mulcli 11211	. . . . 5 ⊢ (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) ∈ ℂ
87	85, 86, 4, 26	divdiri 11967	. . . 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 11956	. . . . 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 11956	. . . . 5 ⊢ ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) / 2) = ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)
90	88, 89	oveq12i 7422	. . . 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 2790	. . 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 11968	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
93		4div2e2 12407	. . . . . 6 ⊢ (4 / 2) = 2
94	93	oveq1i 7420	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
95	92, 94	eqtri 2786	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
96	15, 24, 4, 26	div23i 11968	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
97	93	oveq1i 7420	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
98	96, 97	eqtri 2786	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
99	95, 98	oveq12i 7422	. . 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 2794	. 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 11471	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 1570 ∈ wcel 2143 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 ≤ cle 11239 − cmin 11436 -cneg 11437 / cdiv 11866 2c2 12290 4c4 12292 ↑cexp 14093 abscabs 15281 ℋchba 31271 +_ℎ cva 31272 ·_ℎ csm 31273 norm_ℎcno 31275 −_ℎ cmv 31277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-hfvadd 31352 ax-hvcom 31353 ax-hvass 31354 ax-hv0cl 31355 ax-hvaddid 31356 ax-hfvmul 31357 ax-hvmulid 31358 ax-hvmulass 31359 ax-hvdistr1 31360 ax-hvdistr2 31361 ax-hvmul0 31362 ax-hfi 31431 ax-his1 31434 ax-his2 31435 ax-his3 31436 ax-his4 31437

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-n0 12500 df-z 12587 df-uz 12858 df-rp 13012 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-hnorm 31320 df-hvsub 31323

This theorem is referenced by: (None)

W3C validator