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Theorem ipval2 30691

Description: Expansion of the inner product value ipval 30687. (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 30687	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
7		ax-icn 11074	. . . . . . . . 9 ⊢ i ∈ ℂ
8	1, 2, 3, 4, 5	ipval2lem4 30690	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ i ∈ ℂ) → ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ)
9	7, 8	mpan2 691	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ)
10		mulcl 11099	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) ∈ ℂ) → (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) ∈ ℂ)
11	7, 9, 10	sylancr 587	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) ∈ ℂ)
12		neg1cn 12119	. . . . . . . . 9 ⊢ -1 ∈ ℂ
13	1, 2, 3, 4, 5	ipval2lem4 30690	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -1 ∈ ℂ) → ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2) ∈ ℂ)
14	12, 13	mpan2 691	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2) ∈ ℂ)
15	11, 14	subcld 11481	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) ∈ ℂ)
16		negicn 11370	. . . . . . . . 9 ⊢ -i ∈ ℂ
17	1, 2, 3, 4, 5	ipval2lem4 30690	. . . . . . . . 9 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ -i ∈ ℂ) → ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ)
18	16, 17	mpan2 691	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ)
19		mulcl 11099	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ) → (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
20	7, 18, 19	sylancr 587	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
21	15, 20	negsubd 11487	. . . . . 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 11578	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = -((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
23	22	oveq2d 7370	. . . . . . . 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 11487	. . . . . . . 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 2768	. . . . . . 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 11562	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2) ∈ ℂ) → (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) = -(i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
27	7, 18, 26	sylancr 587	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) = -(i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
28	25, 27	oveq12d 7372	. . . . . 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 11559	. . . . . . . . . 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 1450	. . . . . . . . 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 584	. . . . . . . 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 7369	. . . . . . 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 11513	. . . . . . 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 2768	. . . . . 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 2778	. . . . 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 30611	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (1𝑆𝐵) = 𝐵)
37	36	oveq2d 7370	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(1𝑆𝐵)) = (𝐴𝐺𝐵))
38	37	fveq2d 6834	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(1𝑆𝐵))) = (𝑁‘(𝐴𝐺𝐵)))
39	38	oveq1d 7369	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
40	39	3adant2 1131	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
41	40	oveq2d 7370	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)) = (1 · ((𝑁‘(𝐴𝐺𝐵))↑2)))
42	1, 2, 3, 4, 5	ipval2lem3 30689	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℝ)
43	42	recnd 11149	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℂ)
44	43	mullidd 11139	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺𝐵))↑2)) = ((𝑁‘(𝐴𝐺𝐵))↑2))
45	41, 44	eqtrd 2768	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)) = ((𝑁‘(𝐴𝐺𝐵))↑2))
46	35, 45	oveq12d 7372	. . . 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 12779	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
48		df-4 12199	. . . . . 6 ⊢ 4 = (3 + 1)
49		oveq2 7362	. . . . . . . 8 ⊢ (𝑘 = 4 → (i↑𝑘) = (i↑4))
50		i4 14115	. . . . . . . 8 ⊢ (i↑4) = 1
51	49, 50	eqtrdi 2784	. . . . . . 7 ⊢ (𝑘 = 4 → (i↑𝑘) = 1)
52	51	oveq1d 7369	. . . . . . . . . 10 ⊢ (𝑘 = 4 → ((i↑𝑘)𝑆𝐵) = (1𝑆𝐵))
53	52	oveq2d 7370	. . . . . . . . 9 ⊢ (𝑘 = 4 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(1𝑆𝐵)))
54	53	fveq2d 6834	. . . . . . . 8 ⊢ (𝑘 = 4 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(1𝑆𝐵))))
55	54	oveq1d 7369	. . . . . . 7 ⊢ (𝑘 = 4 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2))
56	51, 55	oveq12d 7372	. . . . . 6 ⊢ (𝑘 = 4 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (1 · ((𝑁‘(𝐴𝐺(1𝑆𝐵)))↑2)))
57		nnnn0 12397	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
58		expcl 13990	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
59	7, 57, 58	sylancr 587	. . . . . . . 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 30690	. . . . . . . 8 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (i↑𝑘) ∈ ℂ) → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) ∈ ℂ)
62	59, 61	sylan2 593	. . . . . . 7 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) ∈ ℂ)
63	60, 62	mulcld 11141	. . . . . 6 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) ∈ ℂ)
64		df-3 12198	. . . . . . 7 ⊢ 3 = (2 + 1)
65		oveq2 7362	. . . . . . . . 9 ⊢ (𝑘 = 3 → (i↑𝑘) = (i↑3))
66		i3 14114	. . . . . . . . 9 ⊢ (i↑3) = -i
67	65, 66	eqtrdi 2784	. . . . . . . 8 ⊢ (𝑘 = 3 → (i↑𝑘) = -i)
68	67	oveq1d 7369	. . . . . . . . . . 11 ⊢ (𝑘 = 3 → ((i↑𝑘)𝑆𝐵) = (-i𝑆𝐵))
69	68	oveq2d 7370	. . . . . . . . . 10 ⊢ (𝑘 = 3 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(-i𝑆𝐵)))
70	69	fveq2d 6834	. . . . . . . . 9 ⊢ (𝑘 = 3 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(-i𝑆𝐵))))
71	70	oveq1d 7369	. . . . . . . 8 ⊢ (𝑘 = 3 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))
72	67, 71	oveq12d 7372	. . . . . . 7 ⊢ (𝑘 = 3 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (-i · ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))
73		df-2 12197	. . . . . . . 8 ⊢ 2 = (1 + 1)
74		oveq2 7362	. . . . . . . . . 10 ⊢ (𝑘 = 2 → (i↑𝑘) = (i↑2))
75		i2 14113	. . . . . . . . . 10 ⊢ (i↑2) = -1
76	74, 75	eqtrdi 2784	. . . . . . . . 9 ⊢ (𝑘 = 2 → (i↑𝑘) = -1)
77	76	oveq1d 7369	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → ((i↑𝑘)𝑆𝐵) = (-1𝑆𝐵))
78	77	oveq2d 7370	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(-1𝑆𝐵)))
79	78	fveq2d 6834	. . . . . . . . . 10 ⊢ (𝑘 = 2 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
80	79	oveq1d 7369	. . . . . . . . 9 ⊢ (𝑘 = 2 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
81	76, 80	oveq12d 7372	. . . . . . . 8 ⊢ (𝑘 = 2 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (-1 · ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
82		1z 12510	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
83		oveq2 7362	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (i↑𝑘) = (i↑1))
84		exp1 13978	. . . . . . . . . . . . . 14 ⊢ (i ∈ ℂ → (i↑1) = i)
85	7, 84	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (i↑1) = i
86	83, 85	eqtrdi 2784	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (i↑𝑘) = i)
87	86	oveq1d 7369	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((i↑𝑘)𝑆𝐵) = (i𝑆𝐵))
88	87	oveq2d 7370	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (𝐴𝐺((i↑𝑘)𝑆𝐵)) = (𝐴𝐺(i𝑆𝐵)))
89	88	fveq2d 6834	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))) = (𝑁‘(𝐴𝐺(i𝑆𝐵))))
90	89	oveq1d 7369	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2) = ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2))
91	86, 90	oveq12d 7372	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) = (i · ((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2)))
92	91	fsum1 15658	. . . . . . . . . 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 587	. . . . . . . . 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 12145	. . . . . . . . 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 2734	. . . . . . . 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 15680	. . . . . . 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 2734	. . . . . . 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 15680	. . . . . 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 2734	. . . . . 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 15680	. . . . 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 11481	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) ∈ ℂ)
104	9, 18	subcld 11481	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)) ∈ ℂ)
105		mulcl 11099	. . . . . . 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 587	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))) ∈ ℂ)
107	103, 106	addcomd 11324	. . . . 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 11503	. . . . 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 2771	. . . 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 2778	. . 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 7369	. 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 2768	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 1086 = wceq 1541 ∈ wcel 2113 ‘cfv 6488 (class class class)co 7354 ℂcc 11013 1c1 11016 ici 11017 + caddc 11018 · cmul 11020 − cmin 11353 -cneg 11354 / cdiv 11783 ℕcn 12134 2c2 12189 3c3 12190 4c4 12191 ℕ₀cn0 12390 ℤcz 12477 ...cfz 13411 ↑cexp 13972 Σcsu 15597 NrmCVeccnv 30568 +_𝑣 cpv 30569 BaseSetcba 30570 ·_𝑠OLD cns 30571 norm_CVcnmcv 30574 ·_𝑖OLDcdip 30684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-inf2 9540 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092 ax-pre-sup 11093

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7805 df-1st 7929 df-2nd 7930 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-er 8630 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-sup 9335 df-oi 9405 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-div 11784 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-n0 12391 df-z 12478 df-uz 12741 df-rp 12895 df-fz 13412 df-fzo 13559 df-seq 13913 df-exp 13973 df-hash 14242 df-cj 15010 df-re 15011 df-im 15012 df-sqrt 15146 df-abs 15147 df-clim 15399 df-sum 15598 df-grpo 30477 df-ablo 30529 df-vc 30543 df-nv 30576 df-va 30579 df-ba 30580 df-sm 30581 df-0v 30582 df-nmcv 30584 df-dip 30685

This theorem is referenced by: 4ipval2 30692 ipval3 30693 ipidsq 30694 dipcj 30698 dip0r 30701

W3C validator