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Theorem ip0i 30078

Description: A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where 𝐽 is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ip1i.1	⊢ 𝑋 = (BaseSet‘𝑈)
ip1i.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
ip1i.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
ip1i.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)
ip1i.9	⊢ 𝑈 ∈ CPreHil_OLD
ip1i.a	⊢ 𝐴 ∈ 𝑋
ip1i.b	⊢ 𝐵 ∈ 𝑋
ip1i.c	⊢ 𝐶 ∈ 𝑋
ip1i.6	⊢ 𝑁 = (norm_CV‘𝑈)
ip0i.j	⊢ 𝐽 ∈ ℂ

**Assertion**
Ref	Expression
ip0i	⊢ ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2)) + (((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2))) = (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)))

**Proof of Theorem ip0i**
Step	Hyp	Ref	Expression
1		2cn 12287	. . . 4 ⊢ 2 ∈ ℂ
2		ip1i.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
3		ip1i.6	. . . . . . 7 ⊢ 𝑁 = (norm_CV‘𝑈)
4		ip1i.9	. . . . . . . 8 ⊢ 𝑈 ∈ CPreHil_OLD
5	4	phnvi 30069	. . . . . . 7 ⊢ 𝑈 ∈ NrmCVec
6		ip1i.a	. . . . . . . 8 ⊢ 𝐴 ∈ 𝑋
7		ip0i.j	. . . . . . . . 9 ⊢ 𝐽 ∈ ℂ
8		ip1i.c	. . . . . . . . 9 ⊢ 𝐶 ∈ 𝑋
9		ip1i.4	. . . . . . . . . 10 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
10	2, 9	nvscl 29879	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐽 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (𝐽𝑆𝐶) ∈ 𝑋)
11	5, 7, 8, 10	mp3an 1462	. . . . . . . 8 ⊢ (𝐽𝑆𝐶) ∈ 𝑋
12		ip1i.2	. . . . . . . . 9 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
13	2, 12	nvgcl 29873	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋) → (𝐴𝐺(𝐽𝑆𝐶)) ∈ 𝑋)
14	5, 6, 11, 13	mp3an 1462	. . . . . . 7 ⊢ (𝐴𝐺(𝐽𝑆𝐶)) ∈ 𝑋
15	2, 3, 5, 14	nvcli 29915	. . . . . 6 ⊢ (𝑁‘(𝐴𝐺(𝐽𝑆𝐶))) ∈ ℝ
16	15	recni 11228	. . . . 5 ⊢ (𝑁‘(𝐴𝐺(𝐽𝑆𝐶))) ∈ ℂ
17	16	sqcli 14145	. . . 4 ⊢ ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) ∈ ℂ
18	7	negcli 11528	. . . . . . . . 9 ⊢ -𝐽 ∈ ℂ
19	2, 9	nvscl 29879	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ -𝐽 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (-𝐽𝑆𝐶) ∈ 𝑋)
20	5, 18, 8, 19	mp3an 1462	. . . . . . . 8 ⊢ (-𝐽𝑆𝐶) ∈ 𝑋
21	2, 12	nvgcl 29873	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋) → (𝐴𝐺(-𝐽𝑆𝐶)) ∈ 𝑋)
22	5, 6, 20, 21	mp3an 1462	. . . . . . 7 ⊢ (𝐴𝐺(-𝐽𝑆𝐶)) ∈ 𝑋
23	2, 3, 5, 22	nvcli 29915	. . . . . 6 ⊢ (𝑁‘(𝐴𝐺(-𝐽𝑆𝐶))) ∈ ℝ
24	23	recni 11228	. . . . 5 ⊢ (𝑁‘(𝐴𝐺(-𝐽𝑆𝐶))) ∈ ℂ
25	24	sqcli 14145	. . . 4 ⊢ ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2) ∈ ℂ
26	1, 17, 25	subdii 11663	. . 3 ⊢ (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2))) = ((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) − (2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)))
27	1, 17	mulcli 11221	. . . 4 ⊢ (2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) ∈ ℂ
28	1, 25	mulcli 11221	. . . 4 ⊢ (2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) ∈ ℂ
29		ip1i.b	. . . . . . . 8 ⊢ 𝐵 ∈ 𝑋
30	2, 3, 5, 29	nvcli 29915	. . . . . . 7 ⊢ (𝑁‘𝐵) ∈ ℝ
31	30	recni 11228	. . . . . 6 ⊢ (𝑁‘𝐵) ∈ ℂ
32	31	sqcli 14145	. . . . 5 ⊢ ((𝑁‘𝐵)↑2) ∈ ℂ
33	1, 32	mulcli 11221	. . . 4 ⊢ (2 · ((𝑁‘𝐵)↑2)) ∈ ℂ
34		pnpcan2 11500	. . . 4 ⊢ (((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) ∈ ℂ ∧ (2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) ∈ ℂ ∧ (2 · ((𝑁‘𝐵)↑2)) ∈ ℂ) → (((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2))) − ((2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2)))) = ((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) − (2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2))))
35	27, 28, 33, 34	mp3an 1462	. . 3 ⊢ (((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2))) − ((2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2)))) = ((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) − (2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)))
36	26, 35	eqtr4i 2764	. 2 ⊢ (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2))) = (((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2))) − ((2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2))))
37		eqid 2733	. . . . . . . . . 10 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
38	37	nvvc 29868	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
39	12	vafval 29856	. . . . . . . . . 10 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
40	39	vcablo 29822	. . . . . . . . 9 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → 𝐺 ∈ AbelOp)
41	5, 38, 40	mp2b 10	. . . . . . . 8 ⊢ 𝐺 ∈ AbelOp
42	6, 29, 11	3pm3.2i 1340	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋)
43	2, 12	bafval 29857	. . . . . . . . 9 ⊢ 𝑋 = ran 𝐺
44	43	ablo32 29802	. . . . . . . 8 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)) = ((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))
45	41, 42, 44	mp2an 691	. . . . . . 7 ⊢ ((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)) = ((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵)
46	45	fveq2i 6895	. . . . . 6 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶))) = (𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))
47	46	oveq1i 7419	. . . . 5 ⊢ ((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) = ((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))↑2)
48		neg1cn 12326	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
49	2, 9	nvscl 29879	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) ∈ 𝑋)
50	5, 48, 29, 49	mp3an 1462	. . . . . . . . 9 ⊢ (-1𝑆𝐵) ∈ 𝑋
51	6, 50, 11	3pm3.2i 1340	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋)
52	43	ablo32 29802	. . . . . . . 8 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋)) → ((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)) = ((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))
53	41, 51, 52	mp2an 691	. . . . . . 7 ⊢ ((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)) = ((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵))
54	53	fveq2i 6895	. . . . . 6 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶))) = (𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))
55	54	oveq1i 7419	. . . . 5 ⊢ ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) = ((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2)
56	47, 55	oveq12i 7421	. . . 4 ⊢ (((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) + ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2)) = (((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))↑2) + ((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2))
57	2, 12, 9, 3	phpar 30077	. . . . 5 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴𝐺(𝐽𝑆𝐶)) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))↑2) + ((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) + ((𝑁‘𝐵)↑2))))
58	4, 14, 29, 57	mp3an 1462	. . . 4 ⊢ (((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))↑2) + ((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) + ((𝑁‘𝐵)↑2)))
59	1, 17, 32	adddii 11226	. . . 4 ⊢ (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) + ((𝑁‘𝐵)↑2))) = ((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2)))
60	56, 58, 59	3eqtri 2765	. . 3 ⊢ (((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) + ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2)) = ((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2)))
61	6, 29, 20	3pm3.2i 1340	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋)
62	43	ablo32 29802	. . . . . . . 8 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)) = ((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))
63	41, 61, 62	mp2an 691	. . . . . . 7 ⊢ ((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)) = ((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵)
64	63	fveq2i 6895	. . . . . 6 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶))) = (𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))
65	64	oveq1i 7419	. . . . 5 ⊢ ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2) = ((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))↑2)
66	6, 50, 20	3pm3.2i 1340	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋)
67	43	ablo32 29802	. . . . . . . 8 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋)) → ((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)) = ((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))
68	41, 66, 67	mp2an 691	. . . . . . 7 ⊢ ((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)) = ((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵))
69	68	fveq2i 6895	. . . . . 6 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶))) = (𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))
70	69	oveq1i 7419	. . . . 5 ⊢ ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2) = ((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2)
71	65, 70	oveq12i 7421	. . . 4 ⊢ (((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2) + ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2)) = (((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))↑2) + ((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2))
72	2, 12, 9, 3	phpar 30077	. . . . 5 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴𝐺(-𝐽𝑆𝐶)) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))↑2) + ((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2) + ((𝑁‘𝐵)↑2))))
73	4, 22, 29, 72	mp3an 1462	. . . 4 ⊢ (((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))↑2) + ((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2) + ((𝑁‘𝐵)↑2)))
74	1, 25, 32	adddii 11226	. . . 4 ⊢ (2 · (((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2) + ((𝑁‘𝐵)↑2))) = ((2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2)))
75	71, 73, 74	3eqtri 2765	. . 3 ⊢ (((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2) + ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2)) = ((2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2)))
76	60, 75	oveq12i 7421	. 2 ⊢ ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) + ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2)) − (((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2) + ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2))) = (((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2))) − ((2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2))))
77	2, 12	nvgcl 29873	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
78	5, 6, 29, 77	mp3an 1462	. . . . . . 7 ⊢ (𝐴𝐺𝐵) ∈ 𝑋
79	2, 12	nvgcl 29873	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)) ∈ 𝑋)
80	5, 78, 11, 79	mp3an 1462	. . . . . 6 ⊢ ((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)) ∈ 𝑋
81	2, 3, 5, 80	nvcli 29915	. . . . 5 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶))) ∈ ℝ
82	81	recni 11228	. . . 4 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶))) ∈ ℂ
83	82	sqcli 14145	. . 3 ⊢ ((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) ∈ ℂ
84	2, 12	nvgcl 29873	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)
85	5, 6, 50, 84	mp3an 1462	. . . . . . 7 ⊢ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋
86	2, 12	nvgcl 29873	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋) → ((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)) ∈ 𝑋)
87	5, 85, 11, 86	mp3an 1462	. . . . . 6 ⊢ ((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)) ∈ 𝑋
88	2, 3, 5, 87	nvcli 29915	. . . . 5 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶))) ∈ ℝ
89	88	recni 11228	. . . 4 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶))) ∈ ℂ
90	89	sqcli 14145	. . 3 ⊢ ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) ∈ ℂ
91	2, 12	nvgcl 29873	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)) ∈ 𝑋)
92	5, 78, 20, 91	mp3an 1462	. . . . . 6 ⊢ ((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)) ∈ 𝑋
93	2, 3, 5, 92	nvcli 29915	. . . . 5 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶))) ∈ ℝ
94	93	recni 11228	. . . 4 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶))) ∈ ℂ
95	94	sqcli 14145	. . 3 ⊢ ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2) ∈ ℂ
96	2, 12	nvgcl 29873	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋) → ((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)) ∈ 𝑋)
97	5, 85, 20, 96	mp3an 1462	. . . . . 6 ⊢ ((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)) ∈ 𝑋
98	2, 3, 5, 97	nvcli 29915	. . . . 5 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶))) ∈ ℝ
99	98	recni 11228	. . . 4 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶))) ∈ ℂ
100	99	sqcli 14145	. . 3 ⊢ ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2) ∈ ℂ
101	83, 90, 95, 100	addsub4i 11556	. 2 ⊢ ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) + ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2)) − (((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2) + ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2))) = ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2)) + (((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2)))
102	36, 76, 101	3eqtr2ri 2768	1 ⊢ ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2)) + (((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2))) = (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)))

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 ℂcc 11108 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11444 -cneg 11445 2c2 12267 ↑cexp 14027 AbelOpcablo 29797 CVec_OLDcvc 29811 NrmCVeccnv 29837 +_𝑣 cpv 29838 BaseSetcba 29839 ·_𝑠OLD cns 29840 norm_CVcnmcv 29843 ·_𝑖OLDcdip 29953 CPreHil_OLDccphlo 30065

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-seq 13967 df-exp 14028 df-grpo 29746 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-nmcv 29853 df-ph 30066

This theorem is referenced by: ip1ilem 30079

W3C validator