Theorem ipdirilem 9829

Description: Lemma for ipdiri 9830.

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
ipdiri.8	|- A e. X
ipdiri.9	|- B e. X
ipdiri.10	|- C e. X

Assertion

Ref	Expression
ipdirilem	|- ((AGB)PC) = ((APC) + (BPC))

Proof of Theorem ipdirilem

Step	Hyp	Ref	Expression
1		2cn 7164	. . . . . . 7 |- 2 e. CC
2		2ne0 7174	. . . . . . 7 |- 2 =/= 0
3	1, 2	recidi 6916	. . . . . 6 |- (2 x. (1 / 2)) = 1
4	3	opreq1i 4892	. . . . 5 |- ((2 x. (1 / 2))S(AGB)) = (1S(AGB))
5		ip1i.9	. . . . . . 7 |- U e. CPreHil
6	5	phnvi 9816	. . . . . 6 |- U e. NrmCVec
7	1, 2	reccli 6902	. . . . . . 7 |- (1 / 2) e. CC
8		ipdiri.8	. . . . . . . 8 |- A e. X
9		ipdiri.9	. . . . . . . 8 |- B e. X
10		ip1i.1	. . . . . . . . 9 |- X = (BaseSet` U)
11		ip1i.2	. . . . . . . . 9 |- G = (+v` U)
12	10, 11	nvgcl 9571	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (AGB) e. X)
13	6, 8, 9, 12	mp3an 1191	. . . . . . 7 |- (AGB) e. X
14	1, 7, 13	3pm3.2i 1048	. . . . . 6 |- (2 e. CC /\ (1 / 2) e. CC /\ (AGB) e. X)
15		ip1i.4	. . . . . . 7 |- S = (.s` U)
16	10, 15	nvsass 9581	. . . . . 6 |- ((U e. NrmCVec /\ (2 e. CC /\ (1 / 2) e. CC /\ (AGB) e. X)) -> ((2 x. (1 / 2))S(AGB)) = (2S((1 / 2)S(AGB))))
17	6, 14, 16	mp2an 761	. . . . 5 |- ((2 x. (1 / 2))S(AGB)) = (2S((1 / 2)S(AGB)))
18	10, 15	nvsid 9580	. . . . . 6 |- ((U e. NrmCVec /\ (AGB) e. X) -> (1S(AGB)) = (AGB))
19	6, 13, 18	mp2an 761	. . . . 5 |- (1S(AGB)) = (AGB)
20	4, 17, 19	3eqtr3i 1918	. . . 4 |- (2S((1 / 2)S(AGB))) = (AGB)
21	20	opreq1i 4892	. . 3 |- ((2S((1 / 2)S(AGB)))PC) = ((AGB)PC)
22		ip1i.7	. . . 4 |- P = (.i` U)
23	10, 15	nvscl 9579	. . . . 5 |- ((U e. NrmCVec /\ (1 / 2) e. CC /\ (AGB) e. X) -> ((1 / 2)S(AGB)) e. X)
24	6, 7, 13, 23	mp3an 1191	. . . 4 |- ((1 / 2)S(AGB)) e. X
25		ipdiri.10	. . . 4 |- C e. X
26	10, 11, 15, 22, 5, 24, 25	ip2i 9828	. . 3 |- ((2S((1 / 2)S(AGB)))PC) = (2 x. (((1 / 2)S(AGB))PC))
27	21, 26	eqtr3i 1910	. 2 |- ((AGB)PC) = (2 x. (((1 / 2)S(AGB))PC))
28		ax1cn 6422	. . . . . . 7 |- 1 e. CC
29	28	negcli 6526	. . . . . 6 |- -u1 e. CC
30	10, 15	nvscl 9579	. . . . . 6 |- ((U e. NrmCVec /\ -u1 e. CC /\ B e. X) -> (-u1SB) e. X)
31	6, 29, 9, 30	mp3an 1191	. . . . 5 |- (-u1SB) e. X
32	10, 11	nvgcl 9571	. . . . 5 |- ((U e. NrmCVec /\ A e. X /\ (-u1SB) e. X) -> (AG(-u1SB)) e. X)
33	6, 8, 31, 32	mp3an 1191	. . . 4 |- (AG(-u1SB)) e. X
34	10, 15	nvscl 9579	. . . 4 |- ((U e. NrmCVec /\ (1 / 2) e. CC /\ (AG(-u1SB)) e. X) -> ((1 / 2)S(AG(-u1SB))) e. X)
35	6, 7, 33, 34	mp3an 1191	. . 3 |- ((1 / 2)S(AG(-u1SB))) e. X
36	10, 11, 15, 22, 5, 24, 35, 25	ip1i 9827	. 2 |- (((((1 / 2)S(AGB))G((1 / 2)S(AG(-u1SB))))PC) + ((((1 / 2)S(AGB))G(-u1S((1 / 2)S(AG(-u1SB)))))PC)) = (2 x. (((1 / 2)S(AGB))PC))
37	7, 13, 33	3pm3.2i 1048	. . . . . 6 |- ((1 / 2) e. CC /\ (AGB) e. X /\ (AG(-u1SB)) e. X)
38	10, 11, 15	nvdi 9583	. . . . . 6 |- ((U e. NrmCVec /\ ((1 / 2) e. CC /\ (AGB) e. X /\ (AG(-u1SB)) e. X)) -> ((1 / 2)S((AGB)G(AG(-u1SB)))) = (((1 / 2)S(AGB))G((1 / 2)S(AG(-u1SB)))))
39	6, 37, 38	mp2an 761	. . . . 5 |- ((1 / 2)S((AGB)G(AG(-u1SB)))) = (((1 / 2)S(AGB))G((1 / 2)S(AG(-u1SB))))
40		eqid 1884	. . . . . . . . . . . 12 |- (1st` U) = (1st` U)
41	40	nvvc 9566	. . . . . . . . . . 11 |- (U e. NrmCVec -> (1st` U) e. CVec)
42	6, 41	ax-mp 7	. . . . . . . . . 10 |- (1st` U) e. CVec
43	11	vafval 9554	. . . . . . . . . . 11 |- G = (1st` (1st` U))
44	43	vcabl 9508	. . . . . . . . . 10 |- ((1st` U) e. CVec -> G e. Abel)
45	42, 44	ax-mp 7	. . . . . . . . 9 |- G e. Abel
46	8, 9	pm3.2i 307	. . . . . . . . 9 |- (A e. X /\ B e. X)
47	8, 31	pm3.2i 307	. . . . . . . . 9 |- (A e. X /\ (-u1SB) e. X)
48	10, 11	bafval 9555	. . . . . . . . . 10 |- X = ran G
49	48	abl4 9413	. . . . . . . . 9 |- ((G e. Abel /\ (A e. X /\ B e. X) /\ (A e. X /\ (-u1SB) e. X)) -> ((AGB)G(AG(-u1SB))) = ((AGA)G(BG(-u1SB))))
50	45, 46, 47, 49	mp3an 1191	. . . . . . . 8 |- ((AGB)G(AG(-u1SB))) = ((AGA)G(BG(-u1SB)))
51	15	smfval 9556	. . . . . . . . . . 11 |- S = (2nd` (1st` U))
52	43, 51, 48	vc2 9506	. . . . . . . . . 10 |- (((1st` U) e. CVec /\ A e. X) -> (AGA) = (2SA))
53	42, 8, 52	mp2an 761	. . . . . . . . 9 |- (AGA) = (2SA)
54		eqid 1884	. . . . . . . . . . 11 |- (0v` U) = (0v` U)
55	10, 11, 15, 54	nvrinv 9605	. . . . . . . . . 10 |- ((U e. NrmCVec /\ B e. X) -> (BG(-u1SB)) = (0v` U))
56	6, 9, 55	mp2an 761	. . . . . . . . 9 |- (BG(-u1SB)) = (0v` U)
57	53, 56	opreq12i 4894	. . . . . . . 8 |- ((AGA)G(BG(-u1SB))) = ((2SA)G(0v` U))
58	10, 15	nvscl 9579	. . . . . . . . . 10 |- ((U e. NrmCVec /\ 2 e. CC /\ A e. X) -> (2SA) e. X)
59	6, 1, 8, 58	mp3an 1191	. . . . . . . . 9 |- (2SA) e. X
60	10, 11, 54	nv0rid 9588	. . . . . . . . 9 |- ((U e. NrmCVec /\ (2SA) e. X) -> ((2SA)G(0v` U)) = (2SA))
61	6, 59, 60	mp2an 761	. . . . . . . 8 |- ((2SA)G(0v` U)) = (2SA)
62	50, 57, 61	3eqtri 1912	. . . . . . 7 |- ((AGB)G(AG(-u1SB))) = (2SA)
63	62	opreq2i 4893	. . . . . 6 |- ((1 / 2)S((AGB)G(AG(-u1SB)))) = ((1 / 2)S(2SA))
64	7, 1, 8	3pm3.2i 1048	. . . . . . 7 |- ((1 / 2) e. CC /\ 2 e. CC /\ A e. X)
65	10, 15	nvsass 9581	. . . . . . 7 |- ((U e. NrmCVec /\ ((1 / 2) e. CC /\ 2 e. CC /\ A e. X)) -> (((1 / 2) x. 2)SA) = ((1 / 2)S(2SA)))
66	6, 64, 65	mp2an 761	. . . . . 6 |- (((1 / 2) x. 2)SA) = ((1 / 2)S(2SA))
67	28, 1, 2	divcan1i 6906	. . . . . . . 8 |- ((1 / 2) x. 2) = 1
68	67	opreq1i 4892	. . . . . . 7 |- (((1 / 2) x. 2)SA) = (1SA)
69	10, 15	nvsid 9580	. . . . . . . 8 |- ((U e. NrmCVec /\ A e. X) -> (1SA) = A)
70	6, 8, 69	mp2an 761	. . . . . . 7 |- (1SA) = A
71	68, 70	eqtri 1908	. . . . . 6 |- (((1 / 2) x. 2)SA) = A
72	63, 66, 71	3eqtr2i 1915	. . . . 5 |- ((1 / 2)S((AGB)G(AG(-u1SB)))) = A
73	39, 72	eqtr3i 1910	. . . 4 |- (((1 / 2)S(AGB))G((1 / 2)S(AG(-u1SB)))) = A
74	73	opreq1i 4892	. . 3 |- ((((1 / 2)S(AGB))G((1 / 2)S(AG(-u1SB))))PC) = (APC)
75	10, 15	nvscl 9579	. . . . . . . . 9 |- ((U e. NrmCVec /\ -u1 e. CC /\ A e. X) -> (-u1SA) e. X)
76	6, 29, 8, 75	mp3an 1191	. . . . . . . 8 |- (-u1SA) e. X
77	10, 11	nvgcl 9571	. . . . . . . 8 |- ((U e. NrmCVec /\ (-u1SA) e. X /\ B e. X) -> ((-u1SA)GB) e. X)
78	6, 76, 9, 77	mp3an 1191	. . . . . . 7 |- ((-u1SA)GB) e. X
79	7, 13, 78	3pm3.2i 1048	. . . . . 6 |- ((1 / 2) e. CC /\ (AGB) e. X /\ ((-u1SA)GB) e. X)
80	10, 11, 15	nvdi 9583	. . . . . 6 |- ((U e. NrmCVec /\ ((1 / 2) e. CC /\ (AGB) e. X /\ ((-u1SA)GB) e. X)) -> ((1 / 2)S((AGB)G((-u1SA)GB))) = (((1 / 2)S(AGB))G((1 / 2)S((-u1SA)GB))))
81	6, 79, 80	mp2an 761	. . . . 5 |- ((1 / 2)S((AGB)G((-u1SA)GB))) = (((1 / 2)S(AGB))G((1 / 2)S((-u1SA)GB)))
82	76, 9	pm3.2i 307	. . . . . . . . 9 |- ((-u1SA) e. X /\ B e. X)
83	48	abl4 9413	. . . . . . . . 9 |- ((G e. Abel /\ (A e. X /\ B e. X) /\ ((-u1SA) e. X /\ B e. X)) -> ((AGB)G((-u1SA)GB)) = ((AG(-u1SA))G(BGB)))
84	45, 46, 82, 83	mp3an 1191	. . . . . . . 8 |- ((AGB)G((-u1SA)GB)) = ((AG(-u1SA))G(BGB))
85	10, 11, 15, 54	nvrinv 9605	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ A e. X) -> (AG(-u1SA)) = (0v` U))
86	6, 8, 85	mp2an 761	. . . . . . . . . 10 |- (AG(-u1SA)) = (0v` U)
87	86	opreq1i 4892	. . . . . . . . 9 |- ((AG(-u1SA))G(BGB)) = ((0v` U)G(BGB))
88	10, 11	nvgcl 9571	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ B e. X /\ B e. X) -> (BGB) e. X)
89	6, 9, 9, 88	mp3an 1191	. . . . . . . . . 10 |- (BGB) e. X
90	10, 11, 54	nv0lid 9589	. . . . . . . . . 10 |- ((U e. NrmCVec /\ (BGB) e. X) -> ((0v` U)G(BGB)) = (BGB))
91	6, 89, 90	mp2an 761	. . . . . . . . 9 |- ((0v` U)G(BGB)) = (BGB)
92	87, 91	eqtri 1908	. . . . . . . 8 |- ((AG(-u1SA))G(BGB)) = (BGB)
93	43, 51, 48	vc2 9506	. . . . . . . . 9 |- (((1st` U) e. CVec /\ B e. X) -> (BGB) = (2SB))
94	42, 9, 93	mp2an 761	. . . . . . . 8 |- (BGB) = (2SB)
95	84, 92, 94	3eqtri 1912	. . . . . . 7 |- ((AGB)G((-u1SA)GB)) = (2SB)
96	95	opreq2i 4893	. . . . . 6 |- ((1 / 2)S((AGB)G((-u1SA)GB))) = ((1 / 2)S(2SB))
97	7, 1, 9	3pm3.2i 1048	. . . . . . . 8 |- ((1 / 2) e. CC /\ 2 e. CC /\ B e. X)
98	10, 15	nvsass 9581	. . . . . . . 8 |- ((U e. NrmCVec /\ ((1 / 2) e. CC /\ 2 e. CC /\ B e. X)) -> (((1 / 2) x. 2)SB) = ((1 / 2)S(2SB)))
99	6, 97, 98	mp2an 761	. . . . . . 7 |- (((1 / 2) x. 2)SB) = ((1 / 2)S(2SB))
100	67	opreq1i 4892	. . . . . . 7 |- (((1 / 2) x. 2)SB) = (1SB)
101	99, 100	eqtr3i 1910	. . . . . 6 |- ((1 / 2)S(2SB)) = (1SB)
102	10, 15	nvsid 9580	. . . . . . 7 |- ((U e. NrmCVec /\ B e. X) -> (1SB) = B)
103	6, 9, 102	mp2an 761	. . . . . 6 |- (1SB) = B
104	96, 101, 103	3eqtrri 1913	. . . . 5 |- B = ((1 / 2)S((AGB)G((-u1SA)GB)))
105	29, 7, 33	3pm3.2i 1048	. . . . . . . 8 |- (-u1 e. CC /\ (1 / 2) e. CC /\ (AG(-u1SB)) e. X)
106	10, 15	nvsass 9581	. . . . . . . 8 |- ((U e. NrmCVec /\ (-u1 e. CC /\ (1 / 2) e. CC /\ (AG(-u1SB)) e. X)) -> ((-u1 x. (1 / 2))S(AG(-u1SB))) = (-u1S((1 / 2)S(AG(-u1SB)))))
107	6, 105, 106	mp2an 761	. . . . . . 7 |- ((-u1 x. (1 / 2))S(AG(-u1SB))) = (-u1S((1 / 2)S(AG(-u1SB))))
108	29, 7	mulcomi 6476	. . . . . . . . 9 |- (-u1 x. (1 / 2)) = ((1 / 2) x. -u1)
109	108	opreq1i 4892	. . . . . . . 8 |- ((-u1 x. (1 / 2))S(AG(-u1SB))) = (((1 / 2) x. -u1)S(AG(-u1SB)))
110	7, 29, 33	3pm3.2i 1048	. . . . . . . . 9 |- ((1 / 2) e. CC /\ -u1 e. CC /\ (AG(-u1SB)) e. X)
111	10, 15	nvsass 9581	. . . . . . . . 9 |- ((U e. NrmCVec /\ ((1 / 2) e. CC /\ -u1 e. CC /\ (AG(-u1SB)) e. X)) -> (((1 / 2) x. -u1)S(AG(-u1SB))) = ((1 / 2)S(-u1S(AG(-u1SB)))))
112	6, 110, 111	mp2an 761	. . . . . . . 8 |- (((1 / 2) x. -u1)S(AG(-u1SB))) = ((1 / 2)S(-u1S(AG(-u1SB))))
113	29, 8, 31	3pm3.2i 1048	. . . . . . . . . . 11 |- (-u1 e. CC /\ A e. X /\ (-u1SB) e. X)
114	10, 11, 15	nvdi 9583	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ (-u1 e. CC /\ A e. X /\ (-u1SB) e. X)) -> (-u1S(AG(-u1SB))) = ((-u1SA)G(-u1S(-u1SB))))
115	6, 113, 114	mp2an 761	. . . . . . . . . 10 |- (-u1S(AG(-u1SB))) = ((-u1SA)G(-u1S(-u1SB)))
116	28, 28	mul2negi 6610	. . . . . . . . . . . . . 14 |- (-u1 x. -u1) = (1 x. 1)
117	28	mulid1i 6485	. . . . . . . . . . . . . 14 |- (1 x. 1) = 1
118	116, 117	eqtri 1908	. . . . . . . . . . . . 13 |- (-u1 x. -u1) = 1
119	118	opreq1i 4892	. . . . . . . . . . . 12 |- ((-u1 x. -u1)SB) = (1SB)
120	29, 29, 9	3pm3.2i 1048	. . . . . . . . . . . . 13 |- (-u1 e. CC /\ -u1 e. CC /\ B e. X)
121	10, 15	nvsass 9581	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ (-u1 e. CC /\ -u1 e. CC /\ B e. X)) -> ((-u1 x. -u1)SB) = (-u1S(-u1SB)))
122	6, 120, 121	mp2an 761	. . . . . . . . . . . 12 |- ((-u1 x. -u1)SB) = (-u1S(-u1SB))
123	119, 122, 103	3eqtr3i 1918	. . . . . . . . . . 11 |- (-u1S(-u1SB)) = B
124	123	opreq2i 4893	. . . . . . . . . 10 |- ((-u1SA)G(-u1S(-u1SB))) = ((-u1SA)GB)
125	115, 124	eqtri 1908	. . . . . . . . 9 |- (-u1S(AG(-u1SB))) = ((-u1SA)GB)
126	125	opreq2i 4893	. . . . . . . 8 |- ((1 / 2)S(-u1S(AG(-u1SB)))) = ((1 / 2)S((-u1SA)GB))
127	109, 112, 126	3eqtri 1912	. . . . . . 7 |- ((-u1 x. (1 / 2))S(AG(-u1SB))) = ((1 / 2)S((-u1SA)GB))
128	107, 127	eqtr3i 1910	. . . . . 6 |- (-u1S((1 / 2)S(AG(-u1SB)))) = ((1 / 2)S((-u1SA)GB))
129	128	opreq2i 4893	. . . . 5 |- (((1 / 2)S(AGB))G(-u1S((1 / 2)S(AG(-u1SB))))) = (((1 / 2)S(AGB))G((1 / 2)S((-u1SA)GB)))
130	81, 104, 129	3eqtr4ri 1923	. . . 4 |- (((1 / 2)S(AGB))G(-u1S((1 / 2)S(AG(-u1SB))))) = B
131	130	opreq1i 4892	. . 3 |- ((((1 / 2)S(AGB))G(-u1S((1 / 2)S(AG(-u1SB)))))PC) = (BPC)
132	74, 131	opreq12i 4894	. 2 |- (((((1 / 2)S(AGB))G((1 / 2)S(AG(-u1SB))))PC) + ((((1 / 2)S(AGB))G(-u1S((1 / 2)S(AG(-u1SB)))))PC)) = ((APC) + (BPC))
133	27, 36, 132	3eqtr2i 1915	1 |- ((AGB)PC) = ((APC) + (BPC))

Syntax hints:   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  1stc1st 5018  CCcc 6384  1c1 6387   + caddc 6389   x. cmul 6391  -ucneg 6446   / cdiv 6447  2c2 7145  Abelcabl 9407  CVeccvc 9496  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  0vcn0v 9539  .icip 9688  CPreHilcphl 9812

This theorem is referenced by:  ipdiri 9830

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-sup 5664  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-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-sum 8240  df-grp 9316  df-gid 9317  df-ginv 9318  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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