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Theorem normpar2i 31245

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 31107	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
4		2cn 12247	. . . . . . 7 ⊢ 2 ∈ ℂ
5		normpar2.3	. . . . . . 7 ⊢ 𝐶 ∈ ℋ
6	4, 5	hvmulcli 31103	. . . . . 6 ⊢ (2 ·_ℎ 𝐶) ∈ ℋ
7	3, 6	hvsubcli 31110	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
8	7	normcli 31220	. . . 4 ⊢ (norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))) ∈ ℝ
9	8	resqcli 14139	. . 3 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℝ
10	9	recni 11150	. 2 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℂ
11	1, 2	hvsubcli 31110	. . . . 5 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
12	11	normcli 31220	. . . 4 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐵)) ∈ ℝ
13	12	resqcli 14139	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℝ
14	13	recni 11150	. 2 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℂ
15		4cn 12257	. . . . 5 ⊢ 4 ∈ ℂ
16	1, 5	hvsubcli 31110	. . . . . . . 8 ⊢ (𝐴 −_ℎ 𝐶) ∈ ℋ
17	16	normcli 31220	. . . . . . 7 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℝ
18	17	resqcli 14139	. . . . . 6 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℝ
19	18	recni 11150	. . . . 5 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℂ
20	15, 19	mulcli 11143	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) ∈ ℂ
21	2, 5	hvsubcli 31110	. . . . . . . 8 ⊢ (𝐵 −_ℎ 𝐶) ∈ ℋ
22	21	normcli 31220	. . . . . . 7 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℝ
23	22	resqcli 14139	. . . . . 6 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℝ
24	23	recni 11150	. . . . 5 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℂ
25	15, 24	mulcli 11143	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) ∈ ℂ
26		2ne0 12276	. . . 4 ⊢ 2 ≠ 0
27	20, 25, 4, 26	divdiri 11903	. . 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 11328	. . . . . . 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 12135	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℂ
30	29, 6	hvmulcli 31103	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
31	29, 11	hvmulcli 31103	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) ∈ ℋ
32	3, 30, 31	hvadd32i 31143	. . . . . . . . . . . . . 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 31109	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
34	33	oveq1i 7366	. . . . . . . . . . . . . 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 31103	. . . . . . . . . . . . . . . 16 ⊢ (2 ·_ℎ 𝐵) ∈ ℋ
36	35, 6	hvsubvali 31109	. . . . . . . . . . . . . . 15 ⊢ ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
37	1, 2	hvcomi 31108	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 +_ℎ 𝐵) = (𝐵 +_ℎ 𝐴)
38	1, 2	hvnegdii 31151	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) = (𝐵 −_ℎ 𝐴)
39	37, 38	oveq12i 7368	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴))
40	2, 1	hvsubcan2i 31153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴)) = (2 ·_ℎ 𝐵)
41	39, 40	eqtri 2762	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = (2 ·_ℎ 𝐵)
42	41	oveq1i 7366	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 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 31109	. . . . . . . . . . . . 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 31141	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))
47	44, 45, 46	3eqtr4i 2772	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐵 −_ℎ 𝐶))
48	47	fveq2i 6830	. . . . . . . . . . 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 31228	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐵 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
50		0le2 12274	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
51		2re 12246	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
52	51	absidi 15331	. . . . . . . . . . . . 13 ⊢ (0 ≤ 2 → (abs‘2) = 2)
53	50, 52	ax-mp 5	. . . . . . . . . . . 12 ⊢ (abs‘2) = 2
54	53	oveq1i 7366	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
55	48, 49, 54	3eqtri 2766	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
56	55	oveq1i 7366	. . . . . . . . 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 11150	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℂ
58	4, 57	sqmuli 14137	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
59		sq2 14150	. . . . . . . . . 10 ⊢ (2↑2) = 4
60	59	oveq1i 7366	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
61	56, 58, 60	3eqtri 2766	. . . . . . . 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 31153	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ 𝐴)
63	62	oveq1i 7366	. . . . . . . . . . . . . 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 31143	. . . . . . . . . . . . . 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 31103	. . . . . . . . . . . . . . 15 ⊢ (2 ·_ℎ 𝐴) ∈ ℋ
66	65, 6	hvsubvali 31109	. . . . . . . . . . . . . 14 ⊢ ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
67	63, 64, 66	3eqtr4i 2772	. . . . . . . . . . . . 13 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (𝐴 −_ℎ 𝐵)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
68	33	oveq1i 7366	. . . . . . . . . . . . 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 31141	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐴 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
70	67, 68, 69	3eqtr4i 2772	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐴 −_ℎ 𝐶))
71	70	fveq2i 6830	. . . . . . . . . . 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 31228	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐴 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
73	53	oveq1i 7366	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
74	71, 72, 73	3eqtri 2766	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
75	74	oveq1i 7366	. . . . . . . . 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 11150	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℂ
77	4, 76	sqmuli 14137	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
78	59	oveq1i 7366	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
79	75, 77, 78	3eqtri 2766	. . . . . . . 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 7368	. . . . . . 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 2765	. . . . . 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 31243	. . . . . 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 2762	. . . . 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 7366	. . . 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 11143	. . . . 5 ⊢ (2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) ∈ ℂ
86	4, 14	mulcli 11143	. . . . 5 ⊢ (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) ∈ ℂ
87	85, 86, 4, 26	divdiri 11903	. . . 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 11892	. . . . 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 11892	. . . . 5 ⊢ ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) / 2) = ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)
90	88, 89	oveq12i 7368	. . . 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 2766	. . 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 11904	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
93		4div2e2 12337	. . . . . 6 ⊢ (4 / 2) = 2
94	93	oveq1i 7366	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
95	92, 94	eqtri 2762	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
96	15, 24, 4, 26	div23i 11904	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
97	93	oveq1i 7366	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
98	96, 97	eqtri 2762	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
99	95, 98	oveq12i 7368	. . 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 2770	. 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 11403	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 1547 ∈ wcel 2119 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 ≤ cle 11171 − cmin 11368 -cneg 11369 / cdiv 11798 2c2 12227 4c4 12229 ↑cexp 14014 abscabs 15187 ℋchba 31008 +_ℎ cva 31009 ·_ℎ csm 31010 norm_ℎcno 31012 −_ℎ cmv 31014

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-hfvadd 31089 ax-hvcom 31090 ax-hvass 31091 ax-hv0cl 31092 ax-hvaddid 31093 ax-hfvmul 31094 ax-hvmulid 31095 ax-hvmulass 31096 ax-hvdistr1 31097 ax-hvdistr2 31098 ax-hvmul0 31099 ax-hfi 31168 ax-his1 31171 ax-his2 31172 ax-his3 31173 ax-his4 31174

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-hnorm 31057 df-hvsub 31060

This theorem is referenced by: (None)

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