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Theorem ip0i 28860

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 11870	. . . 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 28851	. . . . . . 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 28661	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐽 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (𝐽𝑆𝐶) ∈ 𝑋)
11	5, 7, 8, 10	mp3an 1463	. . . . . . . 8 ⊢ (𝐽𝑆𝐶) ∈ 𝑋
12		ip1i.2	. . . . . . . . 9 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
13	2, 12	nvgcl 28655	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋) → (𝐴𝐺(𝐽𝑆𝐶)) ∈ 𝑋)
14	5, 6, 11, 13	mp3an 1463	. . . . . . 7 ⊢ (𝐴𝐺(𝐽𝑆𝐶)) ∈ 𝑋
15	2, 3, 5, 14	nvcli 28697	. . . . . 6 ⊢ (𝑁‘(𝐴𝐺(𝐽𝑆𝐶))) ∈ ℝ
16	15	recni 10812	. . . . 5 ⊢ (𝑁‘(𝐴𝐺(𝐽𝑆𝐶))) ∈ ℂ
17	16	sqcli 13715	. . . 4 ⊢ ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) ∈ ℂ
18	7	negcli 11111	. . . . . . . . 9 ⊢ -𝐽 ∈ ℂ
19	2, 9	nvscl 28661	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ -𝐽 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (-𝐽𝑆𝐶) ∈ 𝑋)
20	5, 18, 8, 19	mp3an 1463	. . . . . . . 8 ⊢ (-𝐽𝑆𝐶) ∈ 𝑋
21	2, 12	nvgcl 28655	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋) → (𝐴𝐺(-𝐽𝑆𝐶)) ∈ 𝑋)
22	5, 6, 20, 21	mp3an 1463	. . . . . . 7 ⊢ (𝐴𝐺(-𝐽𝑆𝐶)) ∈ 𝑋
23	2, 3, 5, 22	nvcli 28697	. . . . . 6 ⊢ (𝑁‘(𝐴𝐺(-𝐽𝑆𝐶))) ∈ ℝ
24	23	recni 10812	. . . . 5 ⊢ (𝑁‘(𝐴𝐺(-𝐽𝑆𝐶))) ∈ ℂ
25	24	sqcli 13715	. . . 4 ⊢ ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2) ∈ ℂ
26	1, 17, 25	subdii 11246	. . 3 ⊢ (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2))) = ((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) − (2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)))
27	1, 17	mulcli 10805	. . . 4 ⊢ (2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) ∈ ℂ
28	1, 25	mulcli 10805	. . . 4 ⊢ (2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) ∈ ℂ
29		ip1i.b	. . . . . . . 8 ⊢ 𝐵 ∈ 𝑋
30	2, 3, 5, 29	nvcli 28697	. . . . . . 7 ⊢ (𝑁‘𝐵) ∈ ℝ
31	30	recni 10812	. . . . . 6 ⊢ (𝑁‘𝐵) ∈ ℂ
32	31	sqcli 13715	. . . . 5 ⊢ ((𝑁‘𝐵)↑2) ∈ ℂ
33	1, 32	mulcli 10805	. . . 4 ⊢ (2 · ((𝑁‘𝐵)↑2)) ∈ ℂ
34		pnpcan2 11083	. . . 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 1463	. . 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 2762	. 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 2736	. . . . . . . . . 10 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
38	37	nvvc 28650	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
39	12	vafval 28638	. . . . . . . . . 10 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
40	39	vcablo 28604	. . . . . . . . 9 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → 𝐺 ∈ AbelOp)
41	5, 38, 40	mp2b 10	. . . . . . . 8 ⊢ 𝐺 ∈ AbelOp
42	6, 29, 11	3pm3.2i 1341	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋)
43	2, 12	bafval 28639	. . . . . . . . 9 ⊢ 𝑋 = ran 𝐺
44	43	ablo32 28584	. . . . . . . 8 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)) = ((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))
45	41, 42, 44	mp2an 692	. . . . . . 7 ⊢ ((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)) = ((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵)
46	45	fveq2i 6698	. . . . . 6 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶))) = (𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))
47	46	oveq1i 7201	. . . . 5 ⊢ ((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) = ((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺𝐵))↑2)
48		neg1cn 11909	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
49	2, 9	nvscl 28661	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ -1 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) ∈ 𝑋)
50	5, 48, 29, 49	mp3an 1463	. . . . . . . . 9 ⊢ (-1𝑆𝐵) ∈ 𝑋
51	6, 50, 11	3pm3.2i 1341	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋)
52	43	ablo32 28584	. . . . . . . 8 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋)) → ((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)) = ((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))
53	41, 51, 52	mp2an 692	. . . . . . 7 ⊢ ((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)) = ((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵))
54	53	fveq2i 6698	. . . . . 6 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶))) = (𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))
55	54	oveq1i 7201	. . . . 5 ⊢ ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) = ((𝑁‘((𝐴𝐺(𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2)
56	47, 55	oveq12i 7203	. . . 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 28859	. . . . 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 1463	. . . 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 10810	. . . 4 ⊢ (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) + ((𝑁‘𝐵)↑2))) = ((2 · ((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2)))
60	56, 58, 59	3eqtri 2763	. . 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 1341	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋)
62	43	ablo32 28584	. . . . . . . 8 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)) = ((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))
63	41, 61, 62	mp2an 692	. . . . . . 7 ⊢ ((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)) = ((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵)
64	63	fveq2i 6698	. . . . . 6 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶))) = (𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))
65	64	oveq1i 7201	. . . . 5 ⊢ ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2) = ((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺𝐵))↑2)
66	6, 50, 20	3pm3.2i 1341	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋)
67	43	ablo32 28584	. . . . . . . 8 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋)) → ((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)) = ((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))
68	41, 66, 67	mp2an 692	. . . . . . 7 ⊢ ((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)) = ((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵))
69	68	fveq2i 6698	. . . . . 6 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶))) = (𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))
70	69	oveq1i 7201	. . . . 5 ⊢ ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2) = ((𝑁‘((𝐴𝐺(-𝐽𝑆𝐶))𝐺(-1𝑆𝐵)))↑2)
71	65, 70	oveq12i 7203	. . . 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 28859	. . . . 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 1463	. . . 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 10810	. . . 4 ⊢ (2 · (((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2) + ((𝑁‘𝐵)↑2))) = ((2 · ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)) + (2 · ((𝑁‘𝐵)↑2)))
75	71, 73, 74	3eqtri 2763	. . 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 7203	. 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 28655	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
78	5, 6, 29, 77	mp3an 1463	. . . . . . 7 ⊢ (𝐴𝐺𝐵) ∈ 𝑋
79	2, 12	nvgcl 28655	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)) ∈ 𝑋)
80	5, 78, 11, 79	mp3an 1463	. . . . . 6 ⊢ ((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)) ∈ 𝑋
81	2, 3, 5, 80	nvcli 28697	. . . . 5 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶))) ∈ ℝ
82	81	recni 10812	. . . 4 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶))) ∈ ℂ
83	82	sqcli 13715	. . 3 ⊢ ((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) ∈ ℂ
84	2, 12	nvgcl 28655	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ (-1𝑆𝐵) ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋)
85	5, 6, 50, 84	mp3an 1463	. . . . . . 7 ⊢ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋
86	2, 12	nvgcl 28655	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋 ∧ (𝐽𝑆𝐶) ∈ 𝑋) → ((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)) ∈ 𝑋)
87	5, 85, 11, 86	mp3an 1463	. . . . . 6 ⊢ ((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)) ∈ 𝑋
88	2, 3, 5, 87	nvcli 28697	. . . . 5 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶))) ∈ ℝ
89	88	recni 10812	. . . 4 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶))) ∈ ℂ
90	89	sqcli 13715	. . 3 ⊢ ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) ∈ ℂ
91	2, 12	nvgcl 28655	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋) → ((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)) ∈ 𝑋)
92	5, 78, 20, 91	mp3an 1463	. . . . . 6 ⊢ ((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)) ∈ 𝑋
93	2, 3, 5, 92	nvcli 28697	. . . . 5 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶))) ∈ ℝ
94	93	recni 10812	. . . 4 ⊢ (𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶))) ∈ ℂ
95	94	sqcli 13715	. . 3 ⊢ ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2) ∈ ℂ
96	2, 12	nvgcl 28655	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺(-1𝑆𝐵)) ∈ 𝑋 ∧ (-𝐽𝑆𝐶) ∈ 𝑋) → ((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)) ∈ 𝑋)
97	5, 85, 20, 96	mp3an 1463	. . . . . 6 ⊢ ((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)) ∈ 𝑋
98	2, 3, 5, 97	nvcli 28697	. . . . 5 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶))) ∈ ℝ
99	98	recni 10812	. . . 4 ⊢ (𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶))) ∈ ℂ
100	99	sqcli 13715	. . 3 ⊢ ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2) ∈ ℂ
101	83, 90, 95, 100	addsub4i 11139	. 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 2766	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 1089 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 1^st c1st 7737 ℂcc 10692 1c1 10695 + caddc 10697 · cmul 10699 − cmin 11027 -cneg 11028 2c2 11850 ↑cexp 13600 AbelOpcablo 28579 CVec_OLDcvc 28593 NrmCVeccnv 28619 +_𝑣 cpv 28620 BaseSetcba 28621 ·_𝑠OLD cns 28622 norm_CVcnmcv 28625 ·_𝑖OLDcdip 28735 CPreHil_OLDccphlo 28847

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-seq 13540 df-exp 13601 df-grpo 28528 df-ablo 28580 df-vc 28594 df-nv 28627 df-va 28630 df-ba 28631 df-sm 28632 df-0v 28633 df-nmcv 28635 df-ph 28848

This theorem is referenced by: ip1ilem 28861

W3C validator