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Theorem ipval2 30794

Description: Expansion of the inner product value ipval 30790. (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 30790	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
7		ax-icn 11097	. . . . . . . . 9 ⊢ i ∈ ℂ
8	1, 2, 3, 4, 5	ipval2lem4 30793	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ)
9	7, 8	mpan2 692	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ)
10		mulcl 11122	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ) → (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) ∈ ℂ)
11	7, 9, 10	sylancr 588	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) ∈ ℂ)
12		neg1cn 12142	. . . . . . . . 9 ⊢ -1 ∈ ℂ
13	1, 2, 3, 4, 5	ipval2lem4 30793	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2) ∈ ℂ)
14	12, 13	mpan2 692	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2) ∈ ℂ)
15	11, 14	subcld 11504	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) ∈ ℂ)
16		negicn 11393	. . . . . . . . 9 ⊢ -i ∈ ℂ
17	1, 2, 3, 4, 5	ipval2lem4 30793	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ)
18	16, 17	mpan2 692	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ)
19		mulcl 11122	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ) → (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
20	7, 18, 19	sylancr 588	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
21	15, 20	negsubd 11510	. . . . . 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 11601	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = -((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
23	22	oveq2d 7384	. . . . . . . 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 11510	. . . . . . . 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 2772	. . . . . . 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 11585	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ) → (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) = -(i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
27	7, 18, 26	sylancr 588	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) = -(i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
28	25, 27	oveq12d 7386	. . . . . 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 11582	. . . . . . . . . 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 1451	. . . . . . . . 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 585	. . . . . . . 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 7383	. . . . . . 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 11536	. . . . . . 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 2772	. . . . . 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 2782	. . . . 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 30714	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (1𝑆𝐵) = 𝐵)
37	36	oveq2d 7384	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(1𝑆𝐵)) = (𝐴𝐺𝐵))
38	37	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(1𝑆𝐵))) = (𝑁‘(𝐴𝐺𝐵)))
39	38	oveq1d 7383	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
40	39	3adant2 1132	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
41	40	oveq2d 7384	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)) = (1 · ((𝑁‘(𝐴𝐺𝐵))↑2)))
42	1, 2, 3, 4, 5	ipval2lem3 30792	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℝ)
43	42	recnd 11172	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℂ)
44	43	mullidd 11162	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺𝐵))↑2)) = ((𝑁‘(𝐴𝐺𝐵))↑2))
45	41, 44	eqtrd 2772	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)) = ((𝑁‘(𝐴𝐺𝐵))↑2))
46	35, 45	oveq12d 7386	. . . 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 12802	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
48		df-4 12222	. . . . . 6 ⊢ 4 = (3 + 1)
49		oveq2 7376	. . . . . . . 8 ⊢ (𝑘 = 4 → (i↑𝑘) = (i↑4))
50		i4 14139	. . . . . . . 8 ⊢ (i↑4) = 1
51	49, 50	eqtrdi 2788	. . . . . . 7 ⊢ (𝑘 = 4 → (i↑𝑘) = 1)
52	51	oveq1d 7383	. . . . . . . . . 10 ⊢ (𝑘 = 4 → ((i↑𝑘)𝑆𝐵) = (1𝑆𝐵))
53	52	oveq2d 7384	. . . . . . . . 9 ⊢ (𝑘 = 4 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(1𝑆𝐵)))
54	53	fveq2d 6846	. . . . . . . 8 ⊢ (𝑘 = 4 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(1𝑆𝐵))))
55	54	oveq1d 7383	. . . . . . 7 ⊢ (𝑘 = 4 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2))
56	51, 55	oveq12d 7386	. . . . . 6 ⊢ (𝑘 = 4 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)))
57		nnnn0 12420	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
58		expcl 14014	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
59	7, 57, 58	sylancr 588	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (i↑𝑘) ∈ ℂ)
60	59	adantl 481	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (i↑𝑘) ∈ ℂ)
61	1, 2, 3, 4, 5	ipval2lem4 30793	. . . . . . . 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 11164	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) ∈ ℂ)
64		df-3 12221	. . . . . . 7 ⊢ 3 = (2 + 1)
65		oveq2 7376	. . . . . . . . 9 ⊢ (𝑘 = 3 → (i↑𝑘) = (i↑3))
66		i3 14138	. . . . . . . . 9 ⊢ (i↑3) = -i
67	65, 66	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑘 = 3 → (i↑𝑘) = -i)
68	67	oveq1d 7383	. . . . . . . . . . 11 ⊢ (𝑘 = 3 → ((i↑𝑘)𝑆𝐵) = (-i𝑆𝐵))
69	68	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑘 = 3 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(-i𝑆𝐵)))
70	69	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑘 = 3 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(-i𝑆𝐵))))
71	70	oveq1d 7383	. . . . . . . 8 ⊢ (𝑘 = 3 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))
72	67, 71	oveq12d 7386	. . . . . . 7 ⊢ (𝑘 = 3 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
73		df-2 12220	. . . . . . . 8 ⊢ 2 = (1 + 1)
74		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑘 = 2 → (i↑𝑘) = (i↑2))
75		i2 14137	. . . . . . . . . 10 ⊢ (i↑2) = -1
76	74, 75	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝑘 = 2 → (i↑𝑘) = -1)
77	76	oveq1d 7383	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → ((i↑𝑘)𝑆𝐵) = (-1𝑆𝐵))
78	77	oveq2d 7384	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(-1𝑆𝐵)))
79	78	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑘 = 2 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
80	79	oveq1d 7383	. . . . . . . . 9 ⊢ (𝑘 = 2 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
81	76, 80	oveq12d 7386	. . . . . . . 8 ⊢ (𝑘 = 2 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
82		1z 12533	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
83		oveq2 7376	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (i↑𝑘) = (i↑1))
84		exp1 14002	. . . . . . . . . . . . . 14 ⊢ (i ∈ ℂ → (i↑1) = i)
85	7, 84	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (i↑1) = i
86	83, 85	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (i↑𝑘) = i)
87	86	oveq1d 7383	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((i↑𝑘)𝑆𝐵) = (i𝑆𝐵))
88	87	oveq2d 7384	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(i𝑆𝐵)))
89	88	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(i𝑆𝐵))))
90	89	oveq1d 7383	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2))
91	86, 90	oveq12d 7386	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)))
92	91	fsum1 15682	. . . . . . . . . 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 588	. . . . . . . . 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 12168	. . . . . . . . 9 ⊢ 1 ∈ ℕ
95	93, 94	jctil 519	. . . . . . . 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 2738	. . . . . . . 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 15704	. . . . . . 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 2738	. . . . . . 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 15704	. . . . . 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 2738	. . . . . 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 15704	. . . . 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 495	. . . 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 11504	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) ∈ ℂ)
104	9, 18	subcld 11504	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
105		mulcl 11122	. . . . . . 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 588	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) ∈ ℂ)
107	103, 106	addcomd 11347	. . . . 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 11526	. . . . 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 2775	. . . 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 2782	. . 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 7383	. 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 2772	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 1c1 11039 ici 11040 + caddc 11041 · cmul 11043 − cmin 11376 -cneg 11377 / cdiv 11806 ℕcn 12157 2c2 12212 3c3 12213 4c4 12214 ℕ₀cn0 12413 ℤcz 12500 ...cfz 13435 ↑cexp 13996 Σcsu 15621 NrmCVeccnv 30671 +_𝑣 cpv 30672 BaseSetcba 30673 ·_𝑠OLD cns 30674 norm_CVcnmcv 30677 ·_𝑖OLDcdip 30787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-grpo 30580 df-ablo 30632 df-vc 30646 df-nv 30679 df-va 30682 df-ba 30683 df-sm 30684 df-0v 30685 df-nmcv 30687 df-dip 30788

This theorem is referenced by: 4ipval2 30795 ipval3 30796 ipidsq 30797 dipcj 30801 dip0r 30804

W3C validator