Theorem ip1ilem 9826

Description: Lemma for ip1i 9827.

Hypotheses

Ref	Expression
ip1i.1	|- X = (BaseSet` U)
ip1i.2	|- G = (+v` U)
ip1i.4	|- S = (.s` U)
ip1i.7	|- P = (.i` U)
ip1i.9	|- U e. CPreHil
ip1i.a	|- A e. X
ip1i.b	|- B e. X
ip1i.c	|- C e. X
ip1i.6	|- N = (norm` U)
ip0i.j	|- J e. CC

Assertion

Ref	Expression
ip1ilem	|- (((AGB)PC) + ((AG(-u1SB))PC)) = (2 x. (APC))

Proof of Theorem ip1ilem

Step	Hyp	Ref	Expression
1		ip1i.9	. . . . . . 7 |- U e. CPreHil
2	1	phnvi 9816	. . . . . 6 |- U e. NrmCVec
3		ip1i.a	. . . . . 6 |- A e. X
4		ip1i.c	. . . . . 6 |- C e. X
5		ip1i.1	. . . . . . 7 |- X = (BaseSet` U)
6		ip1i.2	. . . . . . 7 |- G = (+v` U)
7		ip1i.4	. . . . . . 7 |- S = (.s` U)
8		ip1i.6	. . . . . . 7 |- N = (norm` U)
9		ip1i.7	. . . . . . 7 |- P = (.i` U)
10	5, 6, 7, 8, 9	4ipval2 9697	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ C e. X) -> (4 x. (APC)) = ((((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)) + (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2)))))
11	2, 3, 4, 10	mp3an 1191	. . . . 5 |- (4 x. (APC)) = ((((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)) + (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2))))
12	11	opreq2i 4893	. . . 4 |- (2 x. (4 x. (APC))) = (2 x. ((((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)) + (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2)))))
13		2cn 7164	. . . . 5 |- 2 e. CC
14		4re 7166	. . . . . 6 |- 4 e. RR
15	14	recni 6467	. . . . 5 |- 4 e. CC
16	5, 9	ipcl 9704	. . . . . 6 |- ((U e. NrmCVec /\ A e. X /\ C e. X) -> (APC) e. CC)
17	2, 3, 4, 16	mp3an 1191	. . . . 5 |- (APC) e. CC
18	13, 15, 17	mul12i 6585	. . . 4 |- (2 x. (4 x. (APC))) = (4 x. (2 x. (APC)))
19	5, 6	nvgcl 9571	. . . . . . . . . . . 12 |- ((U e. NrmCVec /\ A e. X /\ C e. X) -> (AGC) e. X)
20	2, 3, 4, 19	mp3an 1191	. . . . . . . . . . 11 |- (AGC) e. X
21	5, 8, 2, 20	nvcli 9620	. . . . . . . . . 10 |- (N` (AGC)) e. RR
22	21	resqcli 7868	. . . . . . . . 9 |- ((N` (AGC))^2) e. RR
23	22	recni 6467	. . . . . . . 8 |- ((N` (AGC))^2) e. CC
24		ax1cn 6422	. . . . . . . . . . . . . 14 |- 1 e. CC
25	24	negcli 6526	. . . . . . . . . . . . 13 |- -u1 e. CC
26	5, 7	nvscl 9579	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ -u1 e. CC /\ C e. X) -> (-u1SC) e. X)
27	2, 25, 4, 26	mp3an 1191	. . . . . . . . . . . 12 |- (-u1SC) e. X
28	5, 6	nvgcl 9571	. . . . . . . . . . . 12 |- ((U e. NrmCVec /\ A e. X /\ (-u1SC) e. X) -> (AG(-u1SC)) e. X)
29	2, 3, 27, 28	mp3an 1191	. . . . . . . . . . 11 |- (AG(-u1SC)) e. X
30	5, 8, 2, 29	nvcli 9620	. . . . . . . . . 10 |- (N` (AG(-u1SC))) e. RR
31	30	resqcli 7868	. . . . . . . . 9 |- ((N` (AG(-u1SC)))^2) e. RR
32	31	recni 6467	. . . . . . . 8 |- ((N` (AG(-u1SC)))^2) e. CC
33	23, 32	subcli 6523	. . . . . . 7 |- (((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)) e. CC
34		axicn 6423	. . . . . . . 8 |- _i e. CC
35	5, 7	nvscl 9579	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ _i e. CC /\ C e. X) -> (_iSC) e. X)
36	2, 34, 4, 35	mp3an 1191	. . . . . . . . . . . . 13 |- (_iSC) e. X
37	5, 6	nvgcl 9571	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ A e. X /\ (_iSC) e. X) -> (AG(_iSC)) e. X)
38	2, 3, 36, 37	mp3an 1191	. . . . . . . . . . . 12 |- (AG(_iSC)) e. X
39	5, 8, 2, 38	nvcli 9620	. . . . . . . . . . 11 |- (N` (AG(_iSC))) e. RR
40	39	resqcli 7868	. . . . . . . . . 10 |- ((N` (AG(_iSC)))^2) e. RR
41	40	recni 6467	. . . . . . . . 9 |- ((N` (AG(_iSC)))^2) e. CC
42	34	negcli 6526	. . . . . . . . . . . . . 14 |- -u_i e. CC
43	5, 7	nvscl 9579	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ -u_i e. CC /\ C e. X) -> (-u_iSC) e. X)
44	2, 42, 4, 43	mp3an 1191	. . . . . . . . . . . . 13 |- (-u_iSC) e. X
45	5, 6	nvgcl 9571	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ A e. X /\ (-u_iSC) e. X) -> (AG(-u_iSC)) e. X)
46	2, 3, 44, 45	mp3an 1191	. . . . . . . . . . . 12 |- (AG(-u_iSC)) e. X
47	5, 8, 2, 46	nvcli 9620	. . . . . . . . . . 11 |- (N` (AG(-u_iSC))) e. RR
48	47	resqcli 7868	. . . . . . . . . 10 |- ((N` (AG(-u_iSC)))^2) e. RR
49	48	recni 6467	. . . . . . . . 9 |- ((N` (AG(-u_iSC)))^2) e. CC
50	41, 49	subcli 6523	. . . . . . . 8 |- (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2)) e. CC
51	34, 50	mulcli 6474	. . . . . . 7 |- (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2))) e. CC
52	13, 33, 51	adddii 6479	. . . . . 6 |- (2 x. ((((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)) + (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2))))) = ((2 x. (((N` (AGC))^2) - ((N` (AG(-u1SC)))^2))) + (2 x. (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2)))))
53		ip1i.b	. . . . . . . . 9 |- B e. X
54	5, 6, 7, 9, 1, 3, 53, 4, 8, 24	ip0i 9825	. . . . . . . 8 |- ((((N` ((AGB)G(1SC)))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (((N` ((AG(-u1SB))G(1SC)))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2))) = (2 x. (((N` (AG(1SC)))^2) - ((N` (AG(-u1SC)))^2)))
55	5, 7	nvsid 9580	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ C e. X) -> (1SC) = C)
56	2, 4, 55	mp2an 761	. . . . . . . . . . . . 13 |- (1SC) = C
57	56	opreq2i 4893	. . . . . . . . . . . 12 |- ((AGB)G(1SC)) = ((AGB)GC)
58	57	fveq2i 4684	. . . . . . . . . . 11 |- (N` ((AGB)G(1SC))) = (N` ((AGB)GC))
59	58	opreq1i 4892	. . . . . . . . . 10 |- ((N` ((AGB)G(1SC)))^2) = ((N` ((AGB)GC))^2)
60	59	opreq1i 4892	. . . . . . . . 9 |- (((N` ((AGB)G(1SC)))^2) - ((N` ((AGB)G(-u1SC)))^2)) = (((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2))
61	56	opreq2i 4893	. . . . . . . . . . . 12 |- ((AG(-u1SB))G(1SC)) = ((AG(-u1SB))GC)
62	61	fveq2i 4684	. . . . . . . . . . 11 |- (N` ((AG(-u1SB))G(1SC))) = (N` ((AG(-u1SB))GC))
63	62	opreq1i 4892	. . . . . . . . . 10 |- ((N` ((AG(-u1SB))G(1SC)))^2) = ((N` ((AG(-u1SB))GC))^2)
64	63	opreq1i 4892	. . . . . . . . 9 |- (((N` ((AG(-u1SB))G(1SC)))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2)) = (((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2))
65	60, 64	opreq12i 4894	. . . . . . . 8 |- ((((N` ((AGB)G(1SC)))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (((N` ((AG(-u1SB))G(1SC)))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2))) = ((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2)))
66	56	opreq2i 4893	. . . . . . . . . . . 12 |- (AG(1SC)) = (AGC)
67	66	fveq2i 4684	. . . . . . . . . . 11 |- (N` (AG(1SC))) = (N` (AGC))
68	67	opreq1i 4892	. . . . . . . . . 10 |- ((N` (AG(1SC)))^2) = ((N` (AGC))^2)
69	68	opreq1i 4892	. . . . . . . . 9 |- (((N` (AG(1SC)))^2) - ((N` (AG(-u1SC)))^2)) = (((N` (AGC))^2) - ((N` (AG(-u1SC)))^2))
70	69	opreq2i 4893	. . . . . . . 8 |- (2 x. (((N` (AG(1SC)))^2) - ((N` (AG(-u1SC)))^2))) = (2 x. (((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)))
71	54, 65, 70	3eqtr3i 1918	. . . . . . 7 |- ((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2))) = (2 x. (((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)))
72	5, 6, 7, 9, 1, 3, 53, 4, 8, 34	ip0i 9825	. . . . . . . . 9 |- ((((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2)) + (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2))) = (2 x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2)))
73	72	opreq2i 4893	. . . . . . . 8 |- (_i x. ((((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2)) + (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2)))) = (_i x. (2 x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2))))
74	5, 6	nvgcl 9571	. . . . . . . . . . . . . . 15 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (AGB) e. X)
75	2, 3, 53, 74	mp3an 1191	. . . . . . . . . . . . . 14 |- (AGB) e. X
76	5, 6	nvgcl 9571	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ (AGB) e. X /\ (_iSC) e. X) -> ((AGB)G(_iSC)) e. X)
77	2, 75, 36, 76	mp3an 1191	. . . . . . . . . . . . 13 |- ((AGB)G(_iSC)) e. X
78	5, 8, 2, 77	nvcli 9620	. . . . . . . . . . . 12 |- (N` ((AGB)G(_iSC))) e. RR
79	78	resqcli 7868	. . . . . . . . . . 11 |- ((N` ((AGB)G(_iSC)))^2) e. RR
80	79	recni 6467	. . . . . . . . . 10 |- ((N` ((AGB)G(_iSC)))^2) e. CC
81	5, 6	nvgcl 9571	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ (AGB) e. X /\ (-u_iSC) e. X) -> ((AGB)G(-u_iSC)) e. X)
82	2, 75, 44, 81	mp3an 1191	. . . . . . . . . . . . 13 |- ((AGB)G(-u_iSC)) e. X
83	5, 8, 2, 82	nvcli 9620	. . . . . . . . . . . 12 |- (N` ((AGB)G(-u_iSC))) e. RR
84	83	resqcli 7868	. . . . . . . . . . 11 |- ((N` ((AGB)G(-u_iSC)))^2) e. RR
85	84	recni 6467	. . . . . . . . . 10 |- ((N` ((AGB)G(-u_iSC)))^2) e. CC
86	80, 85	subcli 6523	. . . . . . . . 9 |- (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2)) e. CC
87	5, 7	nvscl 9579	. . . . . . . . . . . . . . . 16 |- ((U e. NrmCVec /\ -u1 e. CC /\ B e. X) -> (-u1SB) e. X)
88	2, 25, 53, 87	mp3an 1191	. . . . . . . . . . . . . . 15 |- (-u1SB) e. X
89	5, 6	nvgcl 9571	. . . . . . . . . . . . . . 15 |- ((U e. NrmCVec /\ A e. X /\ (-u1SB) e. X) -> (AG(-u1SB)) e. X)
90	2, 3, 88, 89	mp3an 1191	. . . . . . . . . . . . . 14 |- (AG(-u1SB)) e. X
91	5, 6	nvgcl 9571	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ (AG(-u1SB)) e. X /\ (_iSC) e. X) -> ((AG(-u1SB))G(_iSC)) e. X)
92	2, 90, 36, 91	mp3an 1191	. . . . . . . . . . . . 13 |- ((AG(-u1SB))G(_iSC)) e. X
93	5, 8, 2, 92	nvcli 9620	. . . . . . . . . . . 12 |- (N` ((AG(-u1SB))G(_iSC))) e. RR
94	93	resqcli 7868	. . . . . . . . . . 11 |- ((N` ((AG(-u1SB))G(_iSC)))^2) e. RR
95	94	recni 6467	. . . . . . . . . 10 |- ((N` ((AG(-u1SB))G(_iSC)))^2) e. CC
96	5, 6	nvgcl 9571	. . . . . . . . . . . . . 14 |- ((U e. NrmCVec /\ (AG(-u1SB)) e. X /\ (-u_iSC) e. X) -> ((AG(-u1SB))G(-u_iSC)) e. X)
97	2, 90, 44, 96	mp3an 1191	. . . . . . . . . . . . 13 |- ((AG(-u1SB))G(-u_iSC)) e. X
98	5, 8, 2, 97	nvcli 9620	. . . . . . . . . . . 12 |- (N` ((AG(-u1SB))G(-u_iSC))) e. RR
99	98	resqcli 7868	. . . . . . . . . . 11 |- ((N` ((AG(-u1SB))G(-u_iSC)))^2) e. RR
100	99	recni 6467	. . . . . . . . . 10 |- ((N` ((AG(-u1SB))G(-u_iSC)))^2) e. CC
101	95, 100	subcli 6523	. . . . . . . . 9 |- (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2)) e. CC
102	34, 86, 101	adddii 6479	. . . . . . . 8 |- (_i x. ((((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2)) + (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2)))) = ((_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2))) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2))))
103	34, 13, 50	mul12i 6585	. . . . . . . 8 |- (_i x. (2 x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2)))) = (2 x. (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2))))
104	73, 102, 103	3eqtr3i 1918	. . . . . . 7 |- ((_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2))) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2)))) = (2 x. (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2))))
105	71, 104	opreq12i 4894	. . . . . 6 |- (((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2))) + ((_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2))) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2))))) = ((2 x. (((N` (AGC))^2) - ((N` (AG(-u1SC)))^2))) + (2 x. (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2)))))
106	52, 105	eqtr4i 1911	. . . . 5 |- (2 x. ((((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)) + (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2))))) = (((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2))) + ((_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2))) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2)))))
107	5, 6	nvgcl 9571	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ (AGB) e. X /\ C e. X) -> ((AGB)GC) e. X)
108	2, 75, 4, 107	mp3an 1191	. . . . . . . . . 10 |- ((AGB)GC) e. X
109	5, 8, 2, 108	nvcli 9620	. . . . . . . . 9 |- (N` ((AGB)GC)) e. RR
110	109	resqcli 7868	. . . . . . . 8 |- ((N` ((AGB)GC))^2) e. RR
111	110	recni 6467	. . . . . . 7 |- ((N` ((AGB)GC))^2) e. CC
112	5, 6	nvgcl 9571	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ (AGB) e. X /\ (-u1SC) e. X) -> ((AGB)G(-u1SC)) e. X)
113	2, 75, 27, 112	mp3an 1191	. . . . . . . . . 10 |- ((AGB)G(-u1SC)) e. X
114	5, 8, 2, 113	nvcli 9620	. . . . . . . . 9 |- (N` ((AGB)G(-u1SC))) e. RR
115	114	resqcli 7868	. . . . . . . 8 |- ((N` ((AGB)G(-u1SC)))^2) e. RR
116	115	recni 6467	. . . . . . 7 |- ((N` ((AGB)G(-u1SC)))^2) e. CC
117	111, 116	subcli 6523	. . . . . 6 |- (((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) e. CC
118	5, 6	nvgcl 9571	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ (AG(-u1SB)) e. X /\ C e. X) -> ((AG(-u1SB))GC) e. X)
119	2, 90, 4, 118	mp3an 1191	. . . . . . . . . 10 |- ((AG(-u1SB))GC) e. X
120	5, 8, 2, 119	nvcli 9620	. . . . . . . . 9 |- (N` ((AG(-u1SB))GC)) e. RR
121	120	resqcli 7868	. . . . . . . 8 |- ((N` ((AG(-u1SB))GC))^2) e. RR
122	121	recni 6467	. . . . . . 7 |- ((N` ((AG(-u1SB))GC))^2) e. CC
123	5, 6	nvgcl 9571	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ (AG(-u1SB)) e. X /\ (-u1SC) e. X) -> ((AG(-u1SB))G(-u1SC)) e. X)
124	2, 90, 27, 123	mp3an 1191	. . . . . . . . . 10 |- ((AG(-u1SB))G(-u1SC)) e. X
125	5, 8, 2, 124	nvcli 9620	. . . . . . . . 9 |- (N` ((AG(-u1SB))G(-u1SC))) e. RR
126	125	resqcli 7868	. . . . . . . 8 |- ((N` ((AG(-u1SB))G(-u1SC)))^2) e. RR
127	126	recni 6467	. . . . . . 7 |- ((N` ((AG(-u1SB))G(-u1SC)))^2) e. CC
128	122, 127	subcli 6523	. . . . . 6 |- (((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2)) e. CC
129	34, 86	mulcli 6474	. . . . . 6 |- (_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2))) e. CC
130	34, 101	mulcli 6474	. . . . . 6 |- (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2))) e. CC
131	117, 128, 129, 130	add4i 6496	. . . . 5 |- (((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2))) + ((_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2))) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2))))) = (((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2)))) + ((((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2)) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2)))))
132	5, 9	ipcl 9704	. . . . . . . 8 |- ((U e. NrmCVec /\ (AGB) e. X /\ C e. X) -> ((AGB)PC) e. CC)
133	2, 75, 4, 132	mp3an 1191	. . . . . . 7 |- ((AGB)PC) e. CC
134	5, 9	ipcl 9704	. . . . . . . 8 |- ((U e. NrmCVec /\ (AG(-u1SB)) e. X /\ C e. X) -> ((AG(-u1SB))PC) e. CC)
135	2, 90, 4, 134	mp3an 1191	. . . . . . 7 |- ((AG(-u1SB))PC) e. CC
136	15, 133, 135	adddii 6479	. . . . . 6 |- (4 x. (((AGB)PC) + ((AG(-u1SB))PC))) = ((4 x. ((AGB)PC)) + (4 x. ((AG(-u1SB))PC)))
137	5, 6, 7, 8, 9	4ipval2 9697	. . . . . . . 8 |- ((U e. NrmCVec /\ (AGB) e. X /\ C e. X) -> (4 x. ((AGB)PC)) = ((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2)))))
138	2, 75, 4, 137	mp3an 1191	. . . . . . 7 |- (4 x. ((AGB)PC)) = ((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2))))
139	5, 6, 7, 8, 9	4ipval2 9697	. . . . . . . 8 |- ((U e. NrmCVec /\ (AG(-u1SB)) e. X /\ C e. X) -> (4 x. ((AG(-u1SB))PC)) = ((((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2)) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2)))))
140	2, 90, 4, 139	mp3an 1191	. . . . . . 7 |- (4 x. ((AG(-u1SB))PC)) = ((((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2)) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2))))
141	138, 140	opreq12i 4894	. . . . . 6 |- ((4 x. ((AGB)PC)) + (4 x. ((AG(-u1SB))PC))) = (((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2)))) + ((((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2)) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2)))))
142	136, 141	eqtr2i 1909	. . . . 5 |- (((((N` ((AGB)GC))^2) - ((N` ((AGB)G(-u1SC)))^2)) + (_i x. (((N` ((AGB)G(_iSC)))^2) - ((N` ((AGB)G(-u_iSC)))^2)))) + ((((N` ((AG(-u1SB))GC))^2) - ((N` ((AG(-u1SB))G(-u1SC)))^2)) + (_i x. (((N` ((AG(-u1SB))G(_iSC)))^2) - ((N` ((AG(-u1SB))G(-u_iSC)))^2))))) = (4 x. (((AGB)PC) + ((AG(-u1SB))PC)))
143	106, 131, 142	3eqtri 1912	. . . 4 |- (2 x. ((((N` (AGC))^2) - ((N` (AG(-u1SC)))^2)) + (_i x. (((N` (AG(_iSC)))^2) - ((N` (AG(-u_iSC)))^2))))) = (4 x. (((AGB)PC) + ((AG(-u1SB))PC)))
144	12, 18, 143	3eqtr3ri 1920	. . 3 |- (4 x. (((AGB)PC) + ((AG(-u1SB))PC))) = (4 x. (2 x. (APC)))
145	144	opreq1i 4892	. 2 |- ((4 x. (((AGB)PC) + ((AG(-u1SB))PC))) / 4) = ((4 x. (2 x. (APC))) / 4)
146	133, 135	addcli 6473	. . 3 |- (((AGB)PC) + ((AG(-u1SB))PC)) e. CC
147		4pos 7176	. . . 4 |- 0 < 4
148	14, 147	gt0ne0ii 6799	. . 3 |- 4 =/= 0
149	146, 15, 148	divcan3i 6934	. 2 |- ((4 x. (((AGB)PC) + ((AG(-u1SB))PC))) / 4) = (((AGB)PC) + ((AG(-u1SB))PC))
150	13, 17	mulcli 6474	. . 3 |- (2 x. (APC)) e. CC
151	150, 15, 148	divcan3i 6934	. 2 |- ((4 x. (2 x. (APC))) / 4) = (2 x. (APC))
152	145, 149, 151	3eqtr3i 1918	1 |- (((AGB)PC) + ((AG(-u1SB))PC)) = (2 x. (APC))

Syntax hints:   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  CCcc 6384  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  2c2 7145  4c4 7147  ^cexp 7811  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  normcnm 9541  .icip 9688  CPreHilcphl 9812

This theorem is referenced by:  ip1i 9827

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-div 6892  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  df-ph 9813

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