Theorem sm1cnilem 9686

Description: Lemma for sm1cni 9687.

Hypotheses

Ref	Expression
sm1cni.1	|- X = (BaseSet` U)
sm1cni.4	|- S = (.s` U)
sm1cni.7	|- C = (abs o. - )
sm1cni.8	|- D = (IndMet` U)
sm1cni.j	|- J = (Open` C)
sm1cni.k	|- K = (Open` D)
sm1cni.f	|- F = {<.w, v>. | (w e. CC /\ v = (wSA))}
sm1cni.9	|- U e. NrmCVec
sm1cni.a	|- A e. X
sm1cni.2	|- G = (+v` U)
sm1cnilem.6	|- N = (norm` U)

Assertion

Ref	Expression
sm1cnilem	|- F e. (J Cn K)

Distinct variable groups:   w,v,A   v,S,w   v,X,w

Proof of Theorem sm1cnilem

Step	Hyp	Ref	Expression
1		sm1cni.7	. . . 4 |- C = (abs o. - )
2	1	cnmet 9182	. . 3 |- C e. Met
3		sm1cni.9	. . . 4 |- U e. NrmCVec
4		sm1cni.8	. . . . 5 |- D = (IndMet` U)
5	4	imsmet 9656	. . . 4 |- (U e. NrmCVec -> D e. Met)
6	3, 5	ax-mp 7	. . 3 |- D e. Met
7	1	cnmetba 9181	. . . 4 |- CC = dom dom C
8		sm1cni.j	. . . 4 |- J = (Open` C)
9		sm1cni.1	. . . . 5 |- X = (BaseSet` U)
10	9, 4, 3	imsbai 9654	. . . 4 |- X = dom dom D
11		sm1cni.k	. . . 4 |- K = (Open` D)
12	7, 8, 10, 11	metcn 9167	. . 3 |- ((C e. Met /\ D e. Met) -> (F e. (J Cn K) <-> (F:CC-->X /\ A.r e. CC A.s e. RR (0 < s -> E.t e. RR (0 < t /\ A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s))))))
13	2, 6, 12	mp2an 761	. 2 |- (F e. (J Cn K) <-> (F:CC-->X /\ A.r e. CC A.s e. RR (0 < s -> E.t e. RR (0 < t /\ A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s)))))
14		sm1cni.f	. . 3 |- F = {<.w, v>. | (w e. CC /\ v = (wSA))}
15		sm1cni.a	. . . 4 |- A e. X
16		sm1cni.4	. . . . 5 |- S = (.s` U)
17	9, 16	nvscl 9579	. . . 4 |- ((U e. NrmCVec /\ w e. CC /\ A e. X) -> (wSA) e. X)
18	3, 15, 17	mp3an13 1182	. . 3 |- (w e. CC -> (wSA) e. X)
19	14, 18	fopab 4800	. 2 |- F:CC-->X
20		opreq2 4890	. . . . . . . . . . 11 |- ((N` A) = 0 -> ((abs` (r - u)) x. (N` A)) = ((abs` (r - u)) x. 0))
21	20	breq1d 3348	. . . . . . . . . 10 |- ((N` A) = 0 -> (((abs` (r - u)) x. (N` A)) < s <-> ((abs` (r - u)) x. 0) < s))
22	21	imbi2d 674	. . . . . . . . 9 |- ((N` A) = 0 -> (((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s) <-> ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s)))
23	22	ralbidv 2123	. . . . . . . 8 |- ((N` A) = 0 -> (A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s) <-> A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s)))
24	23	anbi2d 678	. . . . . . 7 |- ((N` A) = 0 -> ((0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s)) <-> (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s))))
25	24	rexbidv 2124	. . . . . 6 |- ((N` A) = 0 -> (E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s)) <-> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s))))
26		sm1cnilem.6	. . . . . . . . . . 11 |- N = (norm` U)
27	9, 26, 3, 15	nvcli 9620	. . . . . . . . . 10 |- (N` A) e. RR
28		redivcl 6978	. . . . . . . . . 10 |- ((s e. RR /\ (N` A) e. RR /\ (N` A) =/= 0) -> (s / (N` A)) e. RR)
29	27, 28	mp3an2 1179	. . . . . . . . 9 |- ((s e. RR /\ (N` A) =/= 0) -> (s / (N` A)) e. RR)
30	29	adantll 428	. . . . . . . 8 |- (((r e. CC /\ s e. RR) /\ (N` A) =/= 0) -> (s / (N` A)) e. RR)
31	30	adantlr 429	. . . . . . 7 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) -> (s / (N` A)) e. RR)
32		divgt0 7037	. . . . . . . . . 10 |- (((s e. RR /\ 0 < s) /\ ((N` A) e. RR /\ 0 < (N` A))) -> 0 < (s / (N` A)))
33	27, 32	mpanr1 774	. . . . . . . . 9 |- (((s e. RR /\ 0 < s) /\ 0 < (N` A)) -> 0 < (s / (N` A)))
34	9, 26	nvge0 9634	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ A e. X) -> 0 <_ (N` A))
35	3, 15, 34	mp2an 761	. . . . . . . . . 10 |- 0 <_ (N` A)
36		0re 6603	. . . . . . . . . . . 12 |- 0 e. RR
37	36, 27	ltleni 6751	. . . . . . . . . . 11 |- (0 < (N` A) <-> (0 <_ (N` A) /\ (N` A) =/= 0))
38	37	biimpri 169	. . . . . . . . . 10 |- ((0 <_ (N` A) /\ (N` A) =/= 0) -> 0 < (N` A))
39	35, 38	mpan 759	. . . . . . . . 9 |- ((N` A) =/= 0 -> 0 < (N` A))
40	33, 39	sylan2 500	. . . . . . . 8 |- (((s e. RR /\ 0 < s) /\ (N` A) =/= 0) -> 0 < (s / (N` A)))
41	40	adantlll 432	. . . . . . 7 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) -> 0 < (s / (N` A)))
42		subcl 6524	. . . . . . . . . . . . . 14 |- ((r e. CC /\ u e. CC) -> (r - u) e. CC)
43		abscl 8084	. . . . . . . . . . . . . 14 |- ((r - u) e. CC -> (abs` (r - u)) e. RR)
44	42, 43	syl 12	. . . . . . . . . . . . 13 |- ((r e. CC /\ u e. CC) -> (abs` (r - u)) e. RR)
45	44	adantlr 429	. . . . . . . . . . . 12 |- (((r e. CC /\ s e. RR) /\ u e. CC) -> (abs` (r - u)) e. RR)
46	45	adantlr 429	. . . . . . . . . . 11 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ u e. CC) -> (abs` (r - u)) e. RR)
47	46	adantlr 429	. . . . . . . . . 10 |- (((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) /\ u e. CC) -> (abs` (r - u)) e. RR)
48	27	a1i 8	. . . . . . . . . 10 |- (((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) /\ u e. CC) -> (N` A) e. RR)
49		simplr 449	. . . . . . . . . . 11 |- (((r e. CC /\ s e. RR) /\ 0 < s) -> s e. RR)
50	49	ad2antrr 440	. . . . . . . . . 10 |- (((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) /\ u e. CC) -> s e. RR)
51	39	ad2antlr 441	. . . . . . . . . 10 |- (((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) /\ u e. CC) -> 0 < (N` A))
52		ltmuldivOLD 7046	. . . . . . . . . 10 |- ((((abs` (r - u)) e. RR /\ (N` A) e. RR /\ s e. RR) /\ 0 < (N` A)) -> (((abs` (r - u)) x. (N` A)) < s <-> (abs` (r - u)) < (s / (N` A))))
53	47, 48, 50, 51, 52	syl31anc 1103	. . . . . . . . 9 |- (((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) /\ u e. CC) -> (((abs` (r - u)) x. (N` A)) < s <-> (abs` (r - u)) < (s / (N` A))))
54	53	biimprd 171	. . . . . . . 8 |- (((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) /\ u e. CC) -> ((abs` (r - u)) < (s / (N` A)) -> ((abs` (r - u)) x. (N` A)) < s))
55	54	r19.21aiva 2176	. . . . . . 7 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) -> A.u e. CC ((abs` (r - u)) < (s / (N` A)) -> ((abs` (r - u)) x. (N` A)) < s))
56		breq2 3342	. . . . . . . . 9 |- (t = (s / (N` A)) -> (0 < t <-> 0 < (s / (N` A))))
57		breq2 3342	. . . . . . . . . . 11 |- (t = (s / (N` A)) -> ((abs` (r - u)) < t <-> (abs` (r - u)) < (s / (N` A))))
58	57	imbi1d 675	. . . . . . . . . 10 |- (t = (s / (N` A)) -> (((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s) <-> ((abs` (r - u)) < (s / (N` A)) -> ((abs` (r - u)) x. (N` A)) < s)))
59	58	ralbidv 2123	. . . . . . . . 9 |- (t = (s / (N` A)) -> (A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s) <-> A.u e. CC ((abs` (r - u)) < (s / (N` A)) -> ((abs` (r - u)) x. (N` A)) < s)))
60	56, 59	anbi12d 690	. . . . . . . 8 |- (t = (s / (N` A)) -> ((0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s)) <-> (0 < (s / (N` A)) /\ A.u e. CC ((abs` (r - u)) < (s / (N` A)) -> ((abs` (r - u)) x. (N` A)) < s))))
61	60	rcla4ev 2381	. . . . . . 7 |- (((s / (N` A)) e. RR /\ (0 < (s / (N` A)) /\ A.u e. CC ((abs` (r - u)) < (s / (N` A)) -> ((abs` (r - u)) x. (N` A)) < s))) -> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s)))
62	31, 41, 55, 61	syl12anc 1098	. . . . . 6 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ (N` A) =/= 0) -> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s)))
63	44	recnd 6468	. . . . . . . . . . . . 13 |- ((r e. CC /\ u e. CC) -> (abs` (r - u)) e. CC)
64		mul01 6606	. . . . . . . . . . . . 13 |- ((abs` (r - u)) e. CC -> ((abs` (r - u)) x. 0) = 0)
65	63, 64	syl 12	. . . . . . . . . . . 12 |- ((r e. CC /\ u e. CC) -> ((abs` (r - u)) x. 0) = 0)
66	65	adantlr 429	. . . . . . . . . . 11 |- (((r e. CC /\ s e. RR) /\ u e. CC) -> ((abs` (r - u)) x. 0) = 0)
67	66	adantlr 429	. . . . . . . . . 10 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ u e. CC) -> ((abs` (r - u)) x. 0) = 0)
68		simplr 449	. . . . . . . . . 10 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ u e. CC) -> 0 < s)
69	67, 68	eqbrtrd 3357	. . . . . . . . 9 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ u e. CC) -> ((abs` (r - u)) x. 0) < s)
70	69	a1d 15	. . . . . . . 8 |- ((((r e. CC /\ s e. RR) /\ 0 < s) /\ u e. CC) -> ((abs` (r - u)) < 1 -> ((abs` (r - u)) x. 0) < s))
71	70	r19.21aiva 2176	. . . . . . 7 |- (((r e. CC /\ s e. RR) /\ 0 < s) -> A.u e. CC ((abs` (r - u)) < 1 -> ((abs` (r - u)) x. 0) < s))
72		lt01 6871	. . . . . . . 8 |- 0 < 1
73		1re 6598	. . . . . . . . 9 |- 1 e. RR
74		breq2 3342	. . . . . . . . . . 11 |- (t = 1 -> (0 < t <-> 0 < 1))
75		breq2 3342	. . . . . . . . . . . . 13 |- (t = 1 -> ((abs` (r - u)) < t <-> (abs` (r - u)) < 1))
76	75	imbi1d 675	. . . . . . . . . . . 12 |- (t = 1 -> (((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s) <-> ((abs` (r - u)) < 1 -> ((abs` (r - u)) x. 0) < s)))
77	76	ralbidv 2123	. . . . . . . . . . 11 |- (t = 1 -> (A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s) <-> A.u e. CC ((abs` (r - u)) < 1 -> ((abs` (r - u)) x. 0) < s)))
78	74, 77	anbi12d 690	. . . . . . . . . 10 |- (t = 1 -> ((0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s)) <-> (0 < 1 /\ A.u e. CC ((abs` (r - u)) < 1 -> ((abs` (r - u)) x. 0) < s))))
79	78	rcla4ev 2381	. . . . . . . . 9 |- ((1 e. RR /\ (0 < 1 /\ A.u e. CC ((abs` (r - u)) < 1 -> ((abs` (r - u)) x. 0) < s))) -> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s)))
80	73, 79	mpan 759	. . . . . . . 8 |- ((0 < 1 /\ A.u e. CC ((abs` (r - u)) < 1 -> ((abs` (r - u)) x. 0) < s)) -> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s)))
81	72, 80	mpan 759	. . . . . . 7 |- (A.u e. CC ((abs` (r - u)) < 1 -> ((abs` (r - u)) x. 0) < s) -> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s)))
82	71, 81	syl 12	. . . . . 6 |- (((r e. CC /\ s e. RR) /\ 0 < s) -> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. 0) < s)))
83	25, 62, 82	pm2.61ne 2087	. . . . 5 |- (((r e. CC /\ s e. RR) /\ 0 < s) -> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s)))
84	1	cnmetdval 9180	. . . . . . . . . . 11 |- ((r e. CC /\ u e. CC) -> (rCu) = (abs` (r - u)))
85	84	breq1d 3348	. . . . . . . . . 10 |- ((r e. CC /\ u e. CC) -> ((rCu) < t <-> (abs` (r - u)) < t))
86		opreq1 4889	. . . . . . . . . . . . . 14 |- (w = r -> (wSA) = (rSA))
87		oprex 4907	. . . . . . . . . . . . . 14 |- (rSA) e. _V
88	86, 14, 87	fvopab4 4743	. . . . . . . . . . . . 13 |- (r e. CC -> (F` r) = (rSA))
89		opreq1 4889	. . . . . . . . . . . . . 14 |- (w = u -> (wSA) = (uSA))
90		oprex 4907	. . . . . . . . . . . . . 14 |- (uSA) e. _V
91	89, 14, 90	fvopab4 4743	. . . . . . . . . . . . 13 |- (u e. CC -> (F` u) = (uSA))
92	88, 91	opreqan12d 4902	. . . . . . . . . . . 12 |- ((r e. CC /\ u e. CC) -> ((F` r)D(F` u)) = ((rSA)D(uSA)))
93		sm1cni.2	. . . . . . . . . . . . . . . 16 |- G = (+v` U)
94	9, 93, 16, 26, 4	imsdval2 9650	. . . . . . . . . . . . . . 15 |- ((U e. NrmCVec /\ (rSA) e. X /\ (uSA) e. X) -> ((rSA)D(uSA)) = (N` ((rSA)G(-u1S(uSA)))))
95	3, 94	mp3an1 1178	. . . . . . . . . . . . . 14 |- (((rSA) e. X /\ (uSA) e. X) -> ((rSA)D(uSA)) = (N` ((rSA)G(-u1S(uSA)))))
96	9, 16	nvscl 9579	. . . . . . . . . . . . . . 15 |- ((U e. NrmCVec /\ r e. CC /\ A e. X) -> (rSA) e. X)
97	3, 15, 96	mp3an13 1182	. . . . . . . . . . . . . 14 |- (r e. CC -> (rSA) e. X)
98	9, 16	nvscl 9579	. . . . . . . . . . . . . . 15 |- ((U e. NrmCVec /\ u e. CC /\ A e. X) -> (uSA) e. X)
99	3, 15, 98	mp3an13 1182	. . . . . . . . . . . . . 14 |- (u e. CC -> (uSA) e. X)
100	95, 97, 99	syl2an 503	. . . . . . . . . . . . 13 |- ((r e. CC /\ u e. CC) -> ((rSA)D(uSA)) = (N` ((rSA)G(-u1S(uSA)))))
101		eqid 1884	. . . . . . . . . . . . . . . . . 18 |- (1st` U) = (1st` U)
102	101	nvvc 9566	. . . . . . . . . . . . . . . . 17 |- (U e. NrmCVec -> (1st` U) e. CVec)
103	3, 102	ax-mp 7	. . . . . . . . . . . . . . . 16 |- (1st` U) e. CVec
104	93	vafval 9554	. . . . . . . . . . . . . . . . 17 |- G = (1st` (1st` U))
105	16	smfval 9556	. . . . . . . . . . . . . . . . 17 |- S = (2nd` (1st` U))
106	9, 93	bafval 9555	. . . . . . . . . . . . . . . . 17 |- X = ran G
107	104, 105, 106	vcsubdir 9507	. . . . . . . . . . . . . . . 16 |- (((1st` U) e. CVec /\ (r e. CC /\ u e. CC /\ A e. X)) -> ((r - u)SA) = ((rSA)G(-u1S(uSA))))
108	103, 107	mpan 759	. . . . . . . . . . . . . . 15 |- ((r e. CC /\ u e. CC /\ A e. X) -> ((r - u)SA) = ((rSA)G(-u1S(uSA))))
109	15, 108	mp3an3 1180	. . . . . . . . . . . . . 14 |- ((r e. CC /\ u e. CC) -> ((r - u)SA) = ((rSA)G(-u1S(uSA))))
110	109	fveq2d 4685	. . . . . . . . . . . . 13 |- ((r e. CC /\ u e. CC) -> (N` ((r - u)SA)) = (N` ((rSA)G(-u1S(uSA)))))
111	9, 16, 26	nvs 9622	. . . . . . . . . . . . . . 15 |- ((U e. NrmCVec /\ (r - u) e. CC /\ A e. X) -> (N` ((r - u)SA)) = ((abs` (r - u)) x. (N` A)))
112	3, 15, 111	mp3an13 1182	. . . . . . . . . . . . . 14 |- ((r - u) e. CC -> (N` ((r - u)SA)) = ((abs` (r - u)) x. (N` A)))
113	42, 112	syl 12	. . . . . . . . . . . . 13 |- ((r e. CC /\ u e. CC) -> (N` ((r - u)SA)) = ((abs` (r - u)) x. (N` A)))
114	100, 110, 113	3eqtr2d 1932	. . . . . . . . . . . 12 |- ((r e. CC /\ u e. CC) -> ((rSA)D(uSA)) = ((abs` (r - u)) x. (N` A)))
115	92, 114	eqtrd 1925	. . . . . . . . . . 11 |- ((r e. CC /\ u e. CC) -> ((F` r)D(F` u)) = ((abs` (r - u)) x. (N` A)))
116	115	breq1d 3348	. . . . . . . . . 10 |- ((r e. CC /\ u e. CC) -> (((F` r)D(F` u)) < s <-> ((abs` (r - u)) x. (N` A)) < s))
117	85, 116	imbi12d 688	. . . . . . . . 9 |- ((r e. CC /\ u e. CC) -> (((rCu) < t -> ((F` r)D(F` u)) < s) <-> ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s)))
118	117	ralbidva 2119	. . . . . . . 8 |- (r e. CC -> (A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s) <-> A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s)))
119	118	anbi2d 678	. . . . . . 7 |- (r e. CC -> ((0 < t /\ A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s)) <-> (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s))))
120	119	rexbidv 2124	. . . . . 6 |- (r e. CC -> (E.t e. RR (0 < t /\ A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s)) <-> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s))))
121	120	ad2antrr 440	. . . . 5 |- (((r e. CC /\ s e. RR) /\ 0 < s) -> (E.t e. RR (0 < t /\ A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s)) <-> E.t e. RR (0 < t /\ A.u e. CC ((abs` (r - u)) < t -> ((abs` (r - u)) x. (N` A)) < s))))
122	83, 121	mpbird 213	. . . 4 |- (((r e. CC /\ s e. RR) /\ 0 < s) -> E.t e. RR (0 < t /\ A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s)))
123	122	ex 402	. . 3 |- ((r e. CC /\ s e. RR) -> (0 < s -> E.t e. RR (0 < t /\ A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s))))
124	123	rgen2 2186	. 2 |- A.r e. CC A.s e. RR (0 < s -> E.t e. RR (0 < t /\ A.u e. CC ((rCu) < t -> ((F` r)D(F` u)) < s)))
125	13, 19, 124	mpbir2an 800	1 |- F e. (J Cn K)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   class class class wbr 3338  {copab 3395   o. ccom 3990  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447   <_ cle 6448   < clt 6653  abscabs 8000   Cn ccn 9028  Metcme 9066  Opencopn 9069  CVeccvc 9496  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  normcnm 9541  IndMetcims 9542

This theorem is referenced by:  sm1cni 9687

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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-map 5383  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-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552

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