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Theorem normpar2i 31083

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 30945	. . . . . 6 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
4		2cn 12313	. . . . . . 7 ⊢ 2 ∈ ℂ
5		normpar2.3	. . . . . . 7 ⊢ 𝐶 ∈ ℋ
6	4, 5	hvmulcli 30941	. . . . . 6 ⊢ (2 ·_ℎ 𝐶) ∈ ℋ
7	3, 6	hvsubcli 30948	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
8	7	normcli 31058	. . . 4 ⊢ (norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))) ∈ ℝ
9	8	resqcli 14202	. . 3 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℝ
10	9	recni 11247	. 2 ⊢ ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2) ∈ ℂ
11	1, 2	hvsubcli 30948	. . . . 5 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
12	11	normcli 31058	. . . 4 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐵)) ∈ ℝ
13	12	resqcli 14202	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℝ
14	13	recni 11247	. 2 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) ∈ ℂ
15		4cn 12323	. . . . 5 ⊢ 4 ∈ ℂ
16	1, 5	hvsubcli 30948	. . . . . . . 8 ⊢ (𝐴 −_ℎ 𝐶) ∈ ℋ
17	16	normcli 31058	. . . . . . 7 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℝ
18	17	resqcli 14202	. . . . . 6 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℝ
19	18	recni 11247	. . . . 5 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2) ∈ ℂ
20	15, 19	mulcli 11240	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) ∈ ℂ
21	2, 5	hvsubcli 30948	. . . . . . . 8 ⊢ (𝐵 −_ℎ 𝐶) ∈ ℋ
22	21	normcli 31058	. . . . . . 7 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℝ
23	22	resqcli 14202	. . . . . 6 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℝ
24	23	recni 11247	. . . . 5 ⊢ ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2) ∈ ℂ
25	15, 24	mulcli 11240	. . . 4 ⊢ (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) ∈ ℂ
26		2ne0 12342	. . . 4 ⊢ 2 ≠ 0
27	20, 25, 4, 26	divdiri 11996	. . 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 11424	. . . . . . 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 12352	. . . . . . . . . . . . . . . 16 ⊢ -1 ∈ ℂ
30	29, 6	hvmulcli 30941	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (2 ·_ℎ 𝐶)) ∈ ℋ
31	29, 11	hvmulcli 30941	. . . . . . . . . . . . . . 15 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) ∈ ℋ
32	3, 30, 31	hvadd32i 30981	. . . . . . . . . . . . . 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 30947	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
34	33	oveq1i 7413	. . . . . . . . . . . . . 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 30941	. . . . . . . . . . . . . . . 16 ⊢ (2 ·_ℎ 𝐵) ∈ ℋ
36	35, 6	hvsubvali 30947	. . . . . . . . . . . . . . 15 ⊢ ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
37	1, 2	hvcomi 30946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 +_ℎ 𝐵) = (𝐵 +_ℎ 𝐴)
38	1, 2	hvnegdii 30989	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 ·_ℎ (𝐴 −_ℎ 𝐵)) = (𝐵 −_ℎ 𝐴)
39	37, 38	oveq12i 7415	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴))
40	2, 1	hvsubcan2i 30991	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 +_ℎ 𝐴) +_ℎ (𝐵 −_ℎ 𝐴)) = (2 ·_ℎ 𝐵)
41	39, 40	eqtri 2758	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (𝐴 −_ℎ 𝐵))) = (2 ·_ℎ 𝐵)
42	41	oveq1i 7413	. . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 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 30947	. . . . . . . . . . . . 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 30979	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶))
47	44, 45, 46	3eqtr4i 2768	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐵 −_ℎ 𝐶))
48	47	fveq2i 6878	. . . . . . . . . . 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 31066	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐵 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
50		0le2 12340	. . . . . . . . . . . . 13 ⊢ 0 ≤ 2
51		2re 12312	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
52	51	absidi 15394	. . . . . . . . . . . . 13 ⊢ (0 ≤ 2 → (abs‘2) = 2)
53	50, 52	ax-mp 5	. . . . . . . . . . . 12 ⊢ (abs‘2) = 2
54	53	oveq1i 7413	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐵 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
55	48, 49, 54	3eqtri 2762	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) −_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))
56	55	oveq1i 7413	. . . . . . . . 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 11247	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐵 −_ℎ 𝐶)) ∈ ℂ
58	4, 57	sqmuli 14200	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐵 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
59		sq2 14213	. . . . . . . . . 10 ⊢ (2↑2) = 4
60	59	oveq1i 7413	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
61	56, 58, 60	3eqtri 2762	. . . . . . . 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 30991	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 +_ℎ 𝐵) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ 𝐴)
63	62	oveq1i 7413	. . . . . . . . . . . . . 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 30981	. . . . . . . . . . . . . 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 30941	. . . . . . . . . . . . . . 15 ⊢ (2 ·_ℎ 𝐴) ∈ ℋ
66	65, 6	hvsubvali 30947	. . . . . . . . . . . . . 14 ⊢ ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶)))
67	63, 64, 66	3eqtr4i 2768	. . . . . . . . . . . . 13 ⊢ (((𝐴 +_ℎ 𝐵) +_ℎ (-1 ·_ℎ (2 ·_ℎ 𝐶))) +_ℎ (𝐴 −_ℎ 𝐵)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
68	33	oveq1i 7413	. . . . . . . . . . . . 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 30979	. . . . . . . . . . . . 13 ⊢ (2 ·_ℎ (𝐴 −_ℎ 𝐶)) = ((2 ·_ℎ 𝐴) −_ℎ (2 ·_ℎ 𝐶))
70	67, 68, 69	3eqtr4i 2768	. . . . . . . . . . . 12 ⊢ (((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵)) = (2 ·_ℎ (𝐴 −_ℎ 𝐶))
71	70	fveq2i 6878	. . . . . . . . . . 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 31066	. . . . . . . . . . 11 ⊢ (norm_ℎ‘(2 ·_ℎ (𝐴 −_ℎ 𝐶))) = ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
73	53	oveq1i 7413	. . . . . . . . . . 11 ⊢ ((abs‘2) · (norm_ℎ‘(𝐴 −_ℎ 𝐶))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
74	71, 72, 73	3eqtri 2762	. . . . . . . . . 10 ⊢ (norm_ℎ‘(((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)) +_ℎ (𝐴 −_ℎ 𝐵))) = (2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))
75	74	oveq1i 7413	. . . . . . . . 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 11247	. . . . . . . . . 10 ⊢ (norm_ℎ‘(𝐴 −_ℎ 𝐶)) ∈ ℂ
77	4, 76	sqmuli 14200	. . . . . . . . 9 ⊢ ((2 · (norm_ℎ‘(𝐴 −_ℎ 𝐶)))↑2) = ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
78	59	oveq1i 7413	. . . . . . . . 9 ⊢ ((2↑2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
79	75, 77, 78	3eqtri 2762	. . . . . . . 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 7415	. . . . . . 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 2761	. . . . . 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 31081	. . . . . 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 2758	. . . . 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 7413	. . . 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 11240	. . . . 5 ⊢ (2 · ((norm_ℎ‘((𝐴 +_ℎ 𝐵) −_ℎ (2 ·_ℎ 𝐶)))↑2)) ∈ ℂ
86	4, 14	mulcli 11240	. . . . 5 ⊢ (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) ∈ ℂ
87	85, 86, 4, 26	divdiri 11996	. . . 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 11985	. . . . 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 11985	. . . . 5 ⊢ ((2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)) / 2) = ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2)
90	88, 89	oveq12i 7415	. . . 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 2762	. . 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 11997	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
93		4d2e2 12408	. . . . . 6 ⊢ (4 / 2) = 2
94	93	oveq1i 7413	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
95	92, 94	eqtri 2758	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐴 −_ℎ 𝐶))↑2))
96	15, 24, 4, 26	div23i 11997	. . . . 5 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
97	93	oveq1i 7413	. . . . 5 ⊢ ((4 / 2) · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
98	96, 97	eqtri 2758	. . . 4 ⊢ ((4 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2)) / 2) = (2 · ((norm_ℎ‘(𝐵 −_ℎ 𝐶))↑2))
99	95, 98	oveq12i 7415	. . 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 2766	. 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 11499	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 1540 ∈ wcel 2108 class class class wbr 5119 ‘cfv 6530 (class class class)co 7403 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 ≤ cle 11268 − cmin 11464 -cneg 11465 / cdiv 11892 2c2 12293 4c4 12295 ↑cexp 14077 abscabs 15251 ℋchba 30846 +_ℎ cva 30847 ·_ℎ csm 30848 norm_ℎcno 30850 −_ℎ cmv 30852

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-hfvadd 30927 ax-hvcom 30928 ax-hvass 30929 ax-hv0cl 30930 ax-hvaddid 30931 ax-hfvmul 30932 ax-hvmulid 30933 ax-hvmulass 30934 ax-hvdistr1 30935 ax-hvdistr2 30936 ax-hvmul0 30937 ax-hfi 31006 ax-his1 31009 ax-his2 31010 ax-his3 31011 ax-his4 31012

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9452 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-n0 12500 df-z 12587 df-uz 12851 df-rp 13007 df-seq 14018 df-exp 14078 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-hnorm 30895 df-hvsub 30898

This theorem is referenced by: (None)

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