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Theorem ipval2 28487

Description: Expansion of the inner product value ipval 28483. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dipfval.1	⊢ 𝑋 = (BaseSet‘𝑈)
dipfval.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
dipfval.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
dipfval.6	⊢ 𝑁 = (norm_CV‘𝑈)
dipfval.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)

**Assertion**
Ref	Expression
ipval2	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) / 4))

Proof of Theorem ipval2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dipfval.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
2		dipfval.2	. . 3 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
3		dipfval.4	. . 3 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
4		dipfval.6	. . 3 ⊢ 𝑁 = (norm_CV‘𝑈)
5		dipfval.7	. . 3 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	ipval 28483	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
7		ax-icn 10599	. . . . . . . . 9 ⊢ i ∈ ℂ
8	1, 2, 3, 4, 5	ipval2lem4 28486	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ)
9	7, 8	mpan2 689	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ)
10		mulcl 10624	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ) → (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) ∈ ℂ)
11	7, 9, 10	sylancr 589	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) ∈ ℂ)
12		neg1cn 11754	. . . . . . . . 9 ⊢ -1 ∈ ℂ
13	1, 2, 3, 4, 5	ipval2lem4 28486	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2) ∈ ℂ)
14	12, 13	mpan2 689	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2) ∈ ℂ)
15	11, 14	subcld 11000	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) ∈ ℂ)
16		negicn 10890	. . . . . . . . 9 ⊢ -i ∈ ℂ
17	1, 2, 3, 4, 5	ipval2lem4 28486	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ)
18	16, 17	mpan2 689	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ)
19		mulcl 10624	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ) → (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
20	7, 18, 19	sylancr 589	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
21	15, 20	negsubd 11006	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + -(i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) = (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) − (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))
22	14	mulm1d 11095	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = -((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
23	22	oveq2d 7175	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) = ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + -((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
24	11, 14	negsubd 11006	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + -((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
25	23, 24	eqtrd 2859	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) = ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
26		mulneg1 11079	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ) → (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) = -(i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
27	7, 18, 26	sylancr 589	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) = -(i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
28	25, 27	oveq12d 7177	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) = (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + -(i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))
29		subdi 11076	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ) → (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) = ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))
30	7, 29	mp3an1 1444	. . . . . . . . 9 ⊢ ((((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ) → (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) = ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))
31	9, 18, 30	syl2anc 586	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) = ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))
32	31	oveq1d 7174	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
33	11, 20, 14	sub32d 11032	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) − (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))
34	32, 33	eqtrd 2859	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) − (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))
35	21, 28, 34	3eqtr4d 2869	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) = ((i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
36	1, 3	nvsid 28407	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (1𝑆𝐵) = 𝐵)
37	36	oveq2d 7175	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(1𝑆𝐵)) = (𝐴𝐺𝐵))
38	37	fveq2d 6677	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(1𝑆𝐵))) = (𝑁‘(𝐴𝐺𝐵)))
39	38	oveq1d 7174	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
40	39	3adant2 1127	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
41	40	oveq2d 7175	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)) = (1 · ((𝑁‘(𝐴𝐺𝐵))↑2)))
42	1, 2, 3, 4, 5	ipval2lem3 28485	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℝ)
43	42	recnd 10672	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℂ)
44	43	mulid2d 10662	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺𝐵))↑2)) = ((𝑁‘(𝐴𝐺𝐵))↑2))
45	41, 44	eqtrd 2859	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)) = ((𝑁‘(𝐴𝐺𝐵))↑2))
46	35, 45	oveq12d 7177	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) + (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2))) = (((i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + ((𝑁‘(𝐴𝐺𝐵))↑2)))
47		nnuz 12284	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
48		df-4 11705	. . . . . 6 ⊢ 4 = (3 + 1)
49		oveq2 7167	. . . . . . . 8 ⊢ (𝑘 = 4 → (i↑𝑘) = (i↑4))
50		i4 13570	. . . . . . . 8 ⊢ (i↑4) = 1
51	49, 50	syl6eq 2875	. . . . . . 7 ⊢ (𝑘 = 4 → (i↑𝑘) = 1)
52	51	oveq1d 7174	. . . . . . . . . 10 ⊢ (𝑘 = 4 → ((i↑𝑘)𝑆𝐵) = (1𝑆𝐵))
53	52	oveq2d 7175	. . . . . . . . 9 ⊢ (𝑘 = 4 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(1𝑆𝐵)))
54	53	fveq2d 6677	. . . . . . . 8 ⊢ (𝑘 = 4 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(1𝑆𝐵))))
55	54	oveq1d 7174	. . . . . . 7 ⊢ (𝑘 = 4 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2))
56	51, 55	oveq12d 7177	. . . . . 6 ⊢ (𝑘 = 4 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)))
57		nnnn0 11907	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
58		expcl 13450	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
59	7, 57, 58	sylancr 589	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (i↑𝑘) ∈ ℂ)
60	59	adantl 484	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (i↑𝑘) ∈ ℂ)
61	1, 2, 3, 4, 5	ipval2lem4 28486	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (i↑𝑘) ∈ ℂ) → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) ∈ ℂ)
62	59, 61	sylan2 594	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) ∈ ℂ)
63	60, 62	mulcld 10664	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) ∈ ℂ)
64		df-3 11704	. . . . . . 7 ⊢ 3 = (2 + 1)
65		oveq2 7167	. . . . . . . . 9 ⊢ (𝑘 = 3 → (i↑𝑘) = (i↑3))
66		i3 13569	. . . . . . . . 9 ⊢ (i↑3) = -i
67	65, 66	syl6eq 2875	. . . . . . . 8 ⊢ (𝑘 = 3 → (i↑𝑘) = -i)
68	67	oveq1d 7174	. . . . . . . . . . 11 ⊢ (𝑘 = 3 → ((i↑𝑘)𝑆𝐵) = (-i𝑆𝐵))
69	68	oveq2d 7175	. . . . . . . . . 10 ⊢ (𝑘 = 3 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(-i𝑆𝐵)))
70	69	fveq2d 6677	. . . . . . . . 9 ⊢ (𝑘 = 3 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(-i𝑆𝐵))))
71	70	oveq1d 7174	. . . . . . . 8 ⊢ (𝑘 = 3 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))
72	67, 71	oveq12d 7177	. . . . . . 7 ⊢ (𝑘 = 3 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
73		df-2 11703	. . . . . . . 8 ⊢ 2 = (1 + 1)
74		oveq2 7167	. . . . . . . . . 10 ⊢ (𝑘 = 2 → (i↑𝑘) = (i↑2))
75		i2 13568	. . . . . . . . . 10 ⊢ (i↑2) = -1
76	74, 75	syl6eq 2875	. . . . . . . . 9 ⊢ (𝑘 = 2 → (i↑𝑘) = -1)
77	76	oveq1d 7174	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → ((i↑𝑘)𝑆𝐵) = (-1𝑆𝐵))
78	77	oveq2d 7175	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(-1𝑆𝐵)))
79	78	fveq2d 6677	. . . . . . . . . 10 ⊢ (𝑘 = 2 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
80	79	oveq1d 7174	. . . . . . . . 9 ⊢ (𝑘 = 2 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
81	76, 80	oveq12d 7177	. . . . . . . 8 ⊢ (𝑘 = 2 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
82		1z 12015	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
83		oveq2 7167	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (i↑𝑘) = (i↑1))
84		exp1 13438	. . . . . . . . . . . . . 14 ⊢ (i ∈ ℂ → (i↑1) = i)
85	7, 84	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (i↑1) = i
86	83, 85	syl6eq 2875	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (i↑𝑘) = i)
87	86	oveq1d 7174	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((i↑𝑘)𝑆𝐵) = (i𝑆𝐵))
88	87	oveq2d 7175	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(i𝑆𝐵)))
89	88	fveq2d 6677	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(i𝑆𝐵))))
90	89	oveq1d 7174	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2))
91	86, 90	oveq12d 7177	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)))
92	91	fsum1 15105	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) ∈ ℂ) → Σ𝑘 ∈ (1...1)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)))
93	82, 11, 92	sylancr 589	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → Σ𝑘 ∈ (1...1)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)))
94		1nn 11652	. . . . . . . . 9 ⊢ 1 ∈ ℕ
95	93, 94	jctil 522	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 ∈ ℕ ∧ Σ𝑘 ∈ (1...1)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2))))
96		eqidd 2825	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) = ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))))
97	47, 73, 81, 63, 95, 96	fsump1i 15127	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 ∈ ℕ ∧ Σ𝑘 ∈ (1...2)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))))
98		eqidd 2825	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) = (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))
99	47, 64, 72, 63, 97, 98	fsump1i 15127	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (3 ∈ ℕ ∧ Σ𝑘 ∈ (1...3)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))))
100		eqidd 2825	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) + (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2))) = ((((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) + (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2))))
101	47, 48, 56, 63, 99, 100	fsump1i 15127	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 ∈ ℕ ∧ Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = ((((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) + (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)))))
102	101	simprd 498	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = ((((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) + (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) + (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) + (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2))))
103	43, 14	subcld 11000	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) ∈ ℂ)
104	9, 18	subcld 11000	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
105		mulcl 10624	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ) → (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) ∈ ℂ)
106	7, 104, 105	sylancr 589	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) ∈ ℂ)
107	103, 106	addcomd 10845	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) = ((i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) + (((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))))
108	106, 14, 43	subadd23d 11022	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + ((𝑁‘(𝐴𝐺𝐵))↑2)) = ((i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) + (((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))))
109	107, 108	eqtr4d 2862	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) = (((i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + ((𝑁‘(𝐴𝐺𝐵))↑2)))
110	46, 102, 109	3eqtr4d 2869	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = ((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))))
111	110	oveq1d 7174	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) / 4))
112	6, 111	eqtrd 2859	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) / 4))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 1c1 10541 ici 10542 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 / cdiv 11300 ℕcn 11641 2c2 11695 3c3 11696 4c4 11697 ℕ₀cn0 11900 ℤcz 11984 ...cfz 12895 ↑cexp 13432 Σcsu 15045 NrmCVeccnv 28364 +_𝑣 cpv 28365 BaseSetcba 28366 ·_𝑠OLD cns 28367 norm_CVcnmcv 28370 ·_𝑖OLDcdip 28480

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-grpo 28273 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-nmcv 28380 df-dip 28481

This theorem is referenced by: 4ipval2 28488 ipval3 28489 ipidsq 28490 dipcj 28494 dip0r 28497

W3C validator