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Theorem normpar2i 27397

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 27259	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
4		2cn 10968	. . . . . . 7 ⊢ 2 ∈ ℂ
5		normpar2.3	. . . . . . 7 ⊢ 𝐶 ∈ ℋ
6	4, 5	hvmulcli 27255	. . . . . 6 ⊢ (2 ·_ℎ 𝐶) ∈ ℋ
7	3, 6	hvsubcli 27262	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
8	7	normcli 27372	. . . 4 ⊢ (norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))) ∈ ℝ
9	8	resqcli 12811	. . 3 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℝ
10	9	recni 9931	. 2 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℂ
11	1, 2	hvsubcli 27262	. . . . 5 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
12	11	normcli 27372	. . . 4 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐵)) ∈ ℝ
13	12	resqcli 12811	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℝ
14	13	recni 9931	. 2 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℂ
15		4cn 10975	. . . . 5 ⊢ 4 ∈ ℂ
16	1, 5	hvsubcli 27262	. . . . . . . 8 ⊢ (𝐴 −_ℎ 𝐶) ∈ ℋ
17	16	normcli 27372	. . . . . . 7 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℝ
18	17	resqcli 12811	. . . . . 6 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℝ
19	18	recni 9931	. . . . 5 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℂ
20	15, 19	mulcli 9924	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) ∈ ℂ
21	2, 5	hvsubcli 27262	. . . . . . . 8 ⊢ (𝐵 −_ℎ 𝐶) ∈ ℋ
22	21	normcli 27372	. . . . . . 7 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℝ
23	22	resqcli 12811	. . . . . 6 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℝ
24	23	recni 9931	. . . . 5 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℂ
25	15, 24	mulcli 9924	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) ∈ ℂ
26		2ne0 10990	. . . 4 ⊢ 2 ≠ 0
27	20, 25, 4, 26	divdiri 10661	. . 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 10106	. . . . . . 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 11001	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℂ
30	29, 6	hvmulcli 27255	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
31	29, 11	hvmulcli 27255	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) ∈ ℋ
32	3, 30, 31	hvadd32i 27295	. . . . . . . . . . . . . 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 27261	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
34	33	oveq1i 6559	. . . . . . . . . . . . . 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 27255	. . . . . . . . . . . . . . . 16 ⊢ (2 ·_ℎ 𝐵) ∈ ℋ
36	35, 6	hvsubvali 27261	. . . . . . . . . . . . . . 15 ⊢ ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
37	1, 2	hvcomi 27260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 +_ℎ 𝐵) = (𝐵 +_ℎ 𝐴)
38	1, 2	hvnegdii 27303	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) = (𝐵 −_ℎ 𝐴)
39	37, 38	oveq12i 6561	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴))
40	2, 1	hvsubcan2i 27305	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴)) = (2 ·_ℎ 𝐵)
41	39, 40	eqtri 2632	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = (2 ·_ℎ 𝐵)
42	41	oveq1i 6559	. . . . . . . . . . . . . . 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 2635	. . . . . . . . . . . . . 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 2642	. . . . . . . . . . . . 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 27261	. . . . . . . . . . . . 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 27293	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))
47	44, 45, 46	3eqtr4i 2642	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐵 −_ℎ 𝐶))
48	47	fveq2i 6106	. . . . . . . . . . 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 27380	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐵 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
50		0le2 10988	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
51		2re 10967	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
52	51	absidi 13965	. . . . . . . . . . . . 13 ⊢ (0 ≤ 2 → (abs‘2) = 2)
53	50, 52	ax-mp 5	. . . . . . . . . . . 12 ⊢ (abs‘2) = 2
54	53	oveq1i 6559	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
55	48, 49, 54	3eqtri 2636	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
56	55	oveq1i 6559	. . . . . . . . 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 9931	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℂ
58	4, 57	sqmuli 12809	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
59		sq2 12822	. . . . . . . . . 10 ⊢ (2↑2) = 4
60	59	oveq1i 6559	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
61	56, 58, 60	3eqtri 2636	. . . . . . . 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 27305	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ 𝐴)
63	62	oveq1i 6559	. . . . . . . . . . . . . 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 27295	. . . . . . . . . . . . . 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 27255	. . . . . . . . . . . . . . 15 ⊢ (2 ·_ℎ 𝐴) ∈ ℋ
66	65, 6	hvsubvali 27261	. . . . . . . . . . . . . 14 ⊢ ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
67	63, 64, 66	3eqtr4i 2642	. . . . . . . . . . . . 13 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (𝐴 −_ℎ 𝐵)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
68	33	oveq1i 6559	. . . . . . . . . . . . 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 27293	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐴 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
70	67, 68, 69	3eqtr4i 2642	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐴 −_ℎ 𝐶))
71	70	fveq2i 6106	. . . . . . . . . . 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 27380	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐴 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
73	53	oveq1i 6559	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
74	71, 72, 73	3eqtri 2636	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
75	74	oveq1i 6559	. . . . . . . . 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 9931	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℂ
77	4, 76	sqmuli 12809	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
78	59	oveq1i 6559	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
79	75, 77, 78	3eqtri 2636	. . . . . . . 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 6561	. . . . . . 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 2635	. . . . . 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 27395	. . . . . 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 2632	. . . . 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 6559	. . . 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 9924	. . . . 5 ⊢ (2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) ∈ ℂ
86	4, 14	mulcli 9924	. . . . 5 ⊢ (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) ∈ ℂ
87	85, 86, 4, 26	divdiri 10661	. . . 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 10650	. . . . 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 10650	. . . . 5 ⊢ ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) / 2) = ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)
90	88, 89	oveq12i 6561	. . . 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 2636	. . 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 10662	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
93		4d2e2 11061	. . . . . 6 ⊢ (4 / 2) = 2
94	93	oveq1i 6559	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
95	92, 94	eqtri 2632	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
96	15, 24, 4, 26	div23i 10662	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
97	93	oveq1i 6559	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
98	96, 97	eqtri 2632	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
99	95, 98	oveq12i 6561	. . 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 2640	. 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 10178	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 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 4c4 10949 ↑cexp 12722 abscabs 13822 ℋchil 27160 +_ℎ cva 27161 ·_ℎ csm 27162 norm_ℎcno 27164 −_ℎ cmv 27166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-hnorm 27209 df-hvsub 27212

This theorem is referenced by: (None)

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