Theorem ipval2 9696

Description: Expansion of the inner product value ipval 9692. Warning: The HTML proof page is 0.5MB in size.

Hypotheses

Ref	Expression
ipfval.1	|- X = (BaseSet` U)
ipfval.2	|- G = (+v` U)
ipfval.4	|- S = (.s` U)
ipfval.6	|- N = (norm` U)
ipfval.7	|- P = (.i` U)

Assertion

Ref	Expression
ipval2	|- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)))) / 4))

Proof of Theorem ipval2

Step	Hyp	Ref	Expression
1		ipfval.1	. . 3 |- X = (BaseSet` U)
2		ipfval.2	. . 3 |- G = (+v` U)
3		ipfval.4	. . 3 |- S = (.s` U)
4		ipfval.6	. . 3 |- N = (norm` U)
5		ipfval.7	. . 3 |- P = (.i` U)
6	1, 2, 3, 4, 5	ipval 9692	. 2 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (sum_k e. (1...4)((_i^k) x. ((N` (AG((_i^k)SB)))^2)) / 4))
7		axicn 6423	. . . . . . . . . 10 |- _i e. CC
8	1, 2, 3, 4, 5	ipval2lem4 9695	. . . . . . . . . 10 |- (((U e. NrmCVec /\ A e. X /\ B e. X) /\ _i e. CC) -> ((N` (AG(_iSB)))^2) e. CC)
9	7, 8	mpan2 760	. . . . . . . . 9 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((N` (AG(_iSB)))^2) e. CC)
10		mulcl 6456	. . . . . . . . . 10 |- ((_i e. CC /\ ((N` (AG(_iSB)))^2) e. CC) -> (_i x. ((N` (AG(_iSB)))^2)) e. CC)
11	7, 10	mpan 759	. . . . . . . . 9 |- (((N` (AG(_iSB)))^2) e. CC -> (_i x. ((N` (AG(_iSB)))^2)) e. CC)
12	9, 11	syl 12	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (_i x. ((N` (AG(_iSB)))^2)) e. CC)
13		ax1cn 6422	. . . . . . . . . 10 |- 1 e. CC
14	13	negcli 6526	. . . . . . . . 9 |- -u1 e. CC
15	1, 2, 3, 4, 5	ipval2lem4 9695	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. X /\ B e. X) /\ -u1 e. CC) -> ((N` (AG(-u1SB)))^2) e. CC)
16	14, 15	mpan2 760	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((N` (AG(-u1SB)))^2) e. CC)
17		subcl 6524	. . . . . . . 8 |- (((_i x. ((N` (AG(_iSB)))^2)) e. CC /\ ((N` (AG(-u1SB)))^2) e. CC) -> ((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) e. CC)
18	12, 16, 17	syl11anc 524	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) e. CC)
19	7	negcli 6526	. . . . . . . . 9 |- -u_i e. CC
20	1, 2, 3, 4, 5	ipval2lem4 9695	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. X /\ B e. X) /\ -u_i e. CC) -> ((N` (AG(-u_iSB)))^2) e. CC)
21	19, 20	mpan2 760	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((N` (AG(-u_iSB)))^2) e. CC)
22		mulcl 6456	. . . . . . . . 9 |- ((_i e. CC /\ ((N` (AG(-u_iSB)))^2) e. CC) -> (_i x. ((N` (AG(-u_iSB)))^2)) e. CC)
23	7, 22	mpan 759	. . . . . . . 8 |- (((N` (AG(-u_iSB)))^2) e. CC -> (_i x. ((N` (AG(-u_iSB)))^2)) e. CC)
24	21, 23	syl 12	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (_i x. ((N` (AG(-u_iSB)))^2)) e. CC)
25		negsub 6540	. . . . . . 7 |- ((((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) e. CC /\ (_i x. ((N` (AG(-u_iSB)))^2)) e. CC) -> (((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) + -u(_i x. ((N` (AG(-u_iSB)))^2))) = (((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))))
26	18, 24, 25	syl11anc 524	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) + -u(_i x. ((N` (AG(-u_iSB)))^2))) = (((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))))
27		mulm1 6638	. . . . . . . . . 10 |- (((N` (AG(-u1SB)))^2) e. CC -> (-u1 x. ((N` (AG(-u1SB)))^2)) = -u((N` (AG(-u1SB)))^2))
28	16, 27	syl 12	. . . . . . . . 9 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (-u1 x. ((N` (AG(-u1SB)))^2)) = -u((N` (AG(-u1SB)))^2))
29	28	opreq2d 4898	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) = ((_i x. ((N` (AG(_iSB)))^2)) + -u((N` (AG(-u1SB)))^2)))
30		negsub 6540	. . . . . . . . 9 |- (((_i x. ((N` (AG(_iSB)))^2)) e. CC /\ ((N` (AG(-u1SB)))^2) e. CC) -> ((_i x. ((N` (AG(_iSB)))^2)) + -u((N` (AG(-u1SB)))^2)) = ((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)))
31	12, 16, 30	syl11anc 524	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((_i x. ((N` (AG(_iSB)))^2)) + -u((N` (AG(-u1SB)))^2)) = ((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)))
32	29, 31	eqtrd 1925	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) = ((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)))
33		mulneg1 6615	. . . . . . . . 9 |- ((_i e. CC /\ ((N` (AG(-u_iSB)))^2) e. CC) -> (-u_i x. ((N` (AG(-u_iSB)))^2)) = -u(_i x. ((N` (AG(-u_iSB)))^2)))
34	7, 33	mpan 759	. . . . . . . 8 |- (((N` (AG(-u_iSB)))^2) e. CC -> (-u_i x. ((N` (AG(-u_iSB)))^2)) = -u(_i x. ((N` (AG(-u_iSB)))^2)))
35	21, 34	syl 12	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (-u_i x. ((N` (AG(-u_iSB)))^2)) = -u(_i x. ((N` (AG(-u_iSB)))^2)))
36	32, 35	opreq12d 4900	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) + (-u_i x. ((N` (AG(-u_iSB)))^2))) = (((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) + -u(_i x. ((N` (AG(-u_iSB)))^2))))
37		subdi 6590	. . . . . . . . . 10 |- ((_i e. CC /\ ((N` (AG(_iSB)))^2) e. CC /\ ((N` (AG(-u_iSB)))^2) e. CC) -> (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) = ((_i x. ((N` (AG(_iSB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))))
38	7, 37	mp3an1 1178	. . . . . . . . 9 |- ((((N` (AG(_iSB)))^2) e. CC /\ ((N` (AG(-u_iSB)))^2) e. CC) -> (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) = ((_i x. ((N` (AG(_iSB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))))
39	9, 21, 38	syl11anc 524	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) = ((_i x. ((N` (AG(_iSB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))))
40	39	opreq1d 4897	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)) = (((_i x. ((N` (AG(_iSB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)))
41		sub23 6630	. . . . . . . 8 |- (((_i x. ((N` (AG(_iSB)))^2)) e. CC /\ (_i x. ((N` (AG(-u_iSB)))^2)) e. CC /\ ((N` (AG(-u1SB)))^2) e. CC) -> (((_i x. ((N` (AG(_iSB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)) = (((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))))
42	12, 24, 16, 41	syl111anc 1100	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (((_i x. ((N` (AG(_iSB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)) = (((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))))
43	40, 42	eqtrd 1925	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)) = (((_i x. ((N` (AG(_iSB)))^2)) - ((N` (AG(-u1SB)))^2)) - (_i x. ((N` (AG(-u_iSB)))^2))))
44	26, 36, 43	3eqtr4d 1937	. . . . 5 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) + (-u_i x. ((N` (AG(-u_iSB)))^2))) = ((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)))
45		i4 7984	. . . . . . . . 9 |- (_i^4) = 1
46	45	a1i 8	. . . . . . . 8 |- ((U e. NrmCVec /\ B e. X) -> (_i^4) = 1)
47	1, 3	nvsid 9580	. . . . . . . . . . . 12 |- ((U e. NrmCVec /\ B e. X) -> (1SB) = B)
48	45	opreq1i 4892	. . . . . . . . . . . 12 |- ((_i^4)SB) = (1SB)
49	47, 48	syl5eq 1940	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ B e. X) -> ((_i^4)SB) = B)
50	49	opreq2d 4898	. . . . . . . . . 10 |- ((U e. NrmCVec /\ B e. X) -> (AG((_i^4)SB)) = (AGB))
51	50	fveq2d 4685	. . . . . . . . 9 |- ((U e. NrmCVec /\ B e. X) -> (N` (AG((_i^4)SB))) = (N` (AGB)))
52	51	opreq1d 4897	. . . . . . . 8 |- ((U e. NrmCVec /\ B e. X) -> ((N` (AG((_i^4)SB)))^2) = ((N` (AGB))^2))
53	46, 52	opreq12d 4900	. . . . . . 7 |- ((U e. NrmCVec /\ B e. X) -> ((_i^4) x. ((N` (AG((_i^4)SB)))^2)) = (1 x. ((N` (AGB))^2)))
54	53	3adant2 895	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((_i^4) x. ((N` (AG((_i^4)SB)))^2)) = (1 x. ((N` (AGB))^2)))
55	1, 2, 3, 4, 5	ipval2lem3 9694	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((N` (AGB))^2) e. RR)
56	55	recnd 6468	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((N` (AGB))^2) e. CC)
57		mulid2 6578	. . . . . . 7 |- (((N` (AGB))^2) e. CC -> (1 x. ((N` (AGB))^2)) = ((N` (AGB))^2))
58	56, 57	syl 12	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (1 x. ((N` (AGB))^2)) = ((N` (AGB))^2))
59	54, 58	eqtrd 1925	. . . . 5 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((_i^4) x. ((N` (AG((_i^4)SB)))^2)) = ((N` (AGB))^2))
60	44, 59	opreq12d 4900	. . . 4 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) + (-u_i x. ((N` (AG(-u_iSB)))^2))) + ((_i^4) x. ((N` (AG((_i^4)SB)))^2))) = (((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)) + ((N` (AGB))^2)))
61		df-4 7156	. . . . . . . . 9 |- 4 = (3 + 1)
62		3re 7165	. . . . . . . . . . 11 |- 3 e. RR
63	62	recni 6467	. . . . . . . . . 10 |- 3 e. CC
64	63, 13	addcomi 6475	. . . . . . . . 9 |- (3 + 1) = (1 + 3)
65	61, 64	eqtri 1908	. . . . . . . 8 |- 4 = (1 + 3)
66	65	opreq2i 4893	. . . . . . 7 |- (1...4) = (1...(1 + 3))
67	66	sumeq1i 8247	. . . . . 6 |- sum_k e. (1...4)((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = sum_k e. (1...(1 + 3))((_i^k) x. ((N` (AG((_i^k)SB)))^2))
68		1z 7368	. . . . . . 7 |- 1 e. ZZ
69		oprex 4907	. . . . . . . 8 |- ((_i^k) x. ((N` (AG((_i^k)SB)))^2)) e. _V
70	69	fsum4 8285	. . . . . . 7 |- (1 e. ZZ -> sum_k e. (1...(1 + 3))((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((([_1 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) + [_(1 + 1) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) + [_(1 + 2) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) + [_(1 + 3) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))))
71	68, 70	ax-mp 7	. . . . . 6 |- sum_k e. (1...(1 + 3))((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((([_1 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) + [_(1 + 1) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) + [_(1 + 2) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) + [_(1 + 3) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)))
72	68	elisseti 2301	. . . . . . . . . . 11 |- 1 e. _V
73	72	ipval2lem1 9690	. . . . . . . . . 10 |- [_1 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((_i^1) x. ((N` (AG((_i^1)SB)))^2))
74		exp1 7816	. . . . . . . . . . . 12 |- (_i e. CC -> (_i^1) = _i)
75	7, 74	ax-mp 7	. . . . . . . . . . 11 |- (_i^1) = _i
76	75	opreq1i 4892	. . . . . . . . . . . . . 14 |- ((_i^1)SB) = (_iSB)
77	76	opreq2i 4893	. . . . . . . . . . . . 13 |- (AG((_i^1)SB)) = (AG(_iSB))
78	77	fveq2i 4684	. . . . . . . . . . . 12 |- (N` (AG((_i^1)SB))) = (N` (AG(_iSB)))
79	78	opreq1i 4892	. . . . . . . . . . 11 |- ((N` (AG((_i^1)SB)))^2) = ((N` (AG(_iSB)))^2)
80	75, 79	opreq12i 4894	. . . . . . . . . 10 |- ((_i^1) x. ((N` (AG((_i^1)SB)))^2)) = (_i x. ((N` (AG(_iSB)))^2))
81	73, 80	eqtri 1908	. . . . . . . . 9 |- [_1 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = (_i x. ((N` (AG(_iSB)))^2))
82		2cn 7164	. . . . . . . . . . . 12 |- 2 e. CC
83	82	elisseti 2301	. . . . . . . . . . 11 |- 2 e. _V
84	83	ipval2lem1 9690	. . . . . . . . . 10 |- [_2 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((_i^2) x. ((N` (AG((_i^2)SB)))^2))
85		df-2 7154	. . . . . . . . . . 11 |- 2 = (1 + 1)
86		csbeq1 2542	. . . . . . . . . . 11 |- (2 = (1 + 1) -> [_2 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = [_(1 + 1) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)))
87	85, 86	ax-mp 7	. . . . . . . . . 10 |- [_2 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = [_(1 + 1) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))
88		i2 7982	. . . . . . . . . . 11 |- (_i^2) = -u1
89	88	opreq1i 4892	. . . . . . . . . . . . . 14 |- ((_i^2)SB) = (-u1SB)
90	89	opreq2i 4893	. . . . . . . . . . . . 13 |- (AG((_i^2)SB)) = (AG(-u1SB))
91	90	fveq2i 4684	. . . . . . . . . . . 12 |- (N` (AG((_i^2)SB))) = (N` (AG(-u1SB)))
92	91	opreq1i 4892	. . . . . . . . . . 11 |- ((N` (AG((_i^2)SB)))^2) = ((N` (AG(-u1SB)))^2)
93	88, 92	opreq12i 4894	. . . . . . . . . 10 |- ((_i^2) x. ((N` (AG((_i^2)SB)))^2)) = (-u1 x. ((N` (AG(-u1SB)))^2))
94	84, 87, 93	3eqtr3i 1918	. . . . . . . . 9 |- [_(1 + 1) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = (-u1 x. ((N` (AG(-u1SB)))^2))
95	81, 94	opreq12i 4894	. . . . . . . 8 |- ([_1 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) + [_(1 + 1) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) = ((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2)))
96	62	elisseti 2301	. . . . . . . . . 10 |- 3 e. _V
97	96	ipval2lem1 9690	. . . . . . . . 9 |- [_3 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((_i^3) x. ((N` (AG((_i^3)SB)))^2))
98		df-3 7155	. . . . . . . . . . 11 |- 3 = (2 + 1)
99	82, 13	addcomi 6475	. . . . . . . . . . 11 |- (2 + 1) = (1 + 2)
100	98, 99	eqtri 1908	. . . . . . . . . 10 |- 3 = (1 + 2)
101		csbeq1 2542	. . . . . . . . . 10 |- (3 = (1 + 2) -> [_3 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = [_(1 + 2) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)))
102	100, 101	ax-mp 7	. . . . . . . . 9 |- [_3 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = [_(1 + 2) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))
103		i3 7983	. . . . . . . . . 10 |- (_i^3) = -u_i
104	103	opreq1i 4892	. . . . . . . . . . . . 13 |- ((_i^3)SB) = (-u_iSB)
105	104	opreq2i 4893	. . . . . . . . . . . 12 |- (AG((_i^3)SB)) = (AG(-u_iSB))
106	105	fveq2i 4684	. . . . . . . . . . 11 |- (N` (AG((_i^3)SB))) = (N` (AG(-u_iSB)))
107	106	opreq1i 4892	. . . . . . . . . 10 |- ((N` (AG((_i^3)SB)))^2) = ((N` (AG(-u_iSB)))^2)
108	103, 107	opreq12i 4894	. . . . . . . . 9 |- ((_i^3) x. ((N` (AG((_i^3)SB)))^2)) = (-u_i x. ((N` (AG(-u_iSB)))^2))
109	97, 102, 108	3eqtr3i 1918	. . . . . . . 8 |- [_(1 + 2) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = (-u_i x. ((N` (AG(-u_iSB)))^2))
110	95, 109	opreq12i 4894	. . . . . . 7 |- (([_1 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) + [_(1 + 1) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) + [_(1 + 2) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) = (((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) + (-u_i x. ((N` (AG(-u_iSB)))^2)))
111		csbeq1 2542	. . . . . . . . 9 |- (4 = (1 + 3) -> [_4 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = [_(1 + 3) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)))
112	65, 111	ax-mp 7	. . . . . . . 8 |- [_4 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = [_(1 + 3) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))
113		4re 7166	. . . . . . . . . 10 |- 4 e. RR
114	113	elisseti 2301	. . . . . . . . 9 |- 4 e. _V
115	114	ipval2lem1 9690	. . . . . . . 8 |- [_4 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((_i^4) x. ((N` (AG((_i^4)SB)))^2))
116	112, 115	eqtr3i 1910	. . . . . . 7 |- [_(1 + 3) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((_i^4) x. ((N` (AG((_i^4)SB)))^2))
117	110, 116	opreq12i 4894	. . . . . 6 |- ((([_1 / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2)) + [_(1 + 1) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) + [_(1 + 2) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) + [_(1 + 3) / k]_((_i^k) x. ((N` (AG((_i^k)SB)))^2))) = ((((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) + (-u_i x. ((N` (AG(-u_iSB)))^2))) + ((_i^4) x. ((N` (AG((_i^4)SB)))^2)))
118	67, 71, 117	3eqtri 1912	. . . . 5 |- sum_k e. (1...4)((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) + (-u_i x. ((N` (AG(-u_iSB)))^2))) + ((_i^4) x. ((N` (AG((_i^4)SB)))^2)))
119	118	a1i 8	. . . 4 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> sum_k e. (1...4)((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((((_i x. ((N` (AG(_iSB)))^2)) + (-u1 x. ((N` (AG(-u1SB)))^2))) + (-u_i x. ((N` (AG(-u_iSB)))^2))) + ((_i^4) x. ((N` (AG((_i^4)SB)))^2))))
120		subcl 6524	. . . . . . 7 |- ((((N` (AGB))^2) e. CC /\ ((N` (AG(-u1SB)))^2) e. CC) -> (((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) e. CC)
121	56, 16, 120	syl11anc 524	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) e. CC)
122		subcl 6524	. . . . . . . 8 |- ((((N` (AG(_iSB)))^2) e. CC /\ ((N` (AG(-u_iSB)))^2) e. CC) -> (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)) e. CC)
123	9, 21, 122	syl11anc 524	. . . . . . 7 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)) e. CC)
124		mulcl 6456	. . . . . . . 8 |- ((_i e. CC /\ (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)) e. CC) -> (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) e. CC)
125	7, 124	mpan 759	. . . . . . 7 |- ((((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)) e. CC -> (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) e. CC)
126	123, 125	syl 12	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) e. CC)
127		addcom 6458	. . . . . 6 |- (((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) e. CC /\ (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) e. CC) -> ((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)))) = ((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) + (((N` (AGB))^2) - ((N` (AG(-u1SB)))^2))))
128	121, 126, 127	syl11anc 524	. . . . 5 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)))) = ((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) + (((N` (AGB))^2) - ((N` (AG(-u1SB)))^2))))
129		subadd23 6544	. . . . . 6 |- (((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) e. CC /\ ((N` (AG(-u1SB)))^2) e. CC /\ ((N` (AGB))^2) e. CC) -> (((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)) + ((N` (AGB))^2)) = ((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) + (((N` (AGB))^2) - ((N` (AG(-u1SB)))^2))))
130	126, 16, 56, 129	syl111anc 1100	. . . . 5 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)) + ((N` (AGB))^2)) = ((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) + (((N` (AGB))^2) - ((N` (AG(-u1SB)))^2))))
131	128, 130	eqtr4d 1928	. . . 4 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> ((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)))) = (((_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2))) - ((N` (AG(-u1SB)))^2)) + ((N` (AGB))^2)))
132	60, 119, 131	3eqtr4d 1937	. . 3 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> sum_k e. (1...4)((_i^k) x. ((N` (AG((_i^k)SB)))^2)) = ((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)))))
133	132	opreq1d 4897	. 2 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (sum_k e. (1...4)((_i^k) x. ((N` (AG((_i^k)SB)))^2)) / 4) = (((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)))) / 4))
134	6, 133	eqtrd 1925	1 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) = (((((N` (AGB))^2) - ((N` (AG(-u1SB)))^2)) + (_i x. (((N` (AG(_iSB)))^2) - ((N` (AG(-u_iSB)))^2)))) / 4))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  [_csb 2540  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  ZZcz 6451  2c2 7145  3c3 7146  4c4 7147  ...cfz 7637  ^cexp 7811  sum_csu 8239  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  normcnm 9541  .icip 9688

This theorem is referenced by:  4ipval2 9697  ipval3 9698  ipid 9702  ipcj 9706  ip0r 9709

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sum 8240  df-grp 9316  df-gid 9317  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551  df-ip 9689

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