Theorem vacn 9673

Description: Vector addition is continuous. (Contributed by Jeffrey Hankins, 16-Jun-2009.)

Hypotheses

Ref	Expression
vacn.11	|- X = (BaseSet` U)
vacn.12	|- G = (+v` U)
vacn.13	|- C = (IndMet` U)
vacn.14	|- D = {<.<.x, y>., z>. | ((x e. (X X. X) /\ y e. (X X. X)) /\ z = sup({((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
vacn.15	|- J = (Open` C)
vacn.16	|- K = (Open` D)

Assertion

Ref	Expression
vacn	|- (U e. NrmCVec -> G e. (K Cn J))

Distinct variable group:   x,y,z,U

Proof of Theorem vacn

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . 4 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (+v` U) = (+v` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))
2		vacn.12	. . . 4 |- G = (+v` U)
3	1, 2	syl5eq 1940	. . 3 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> G = (+v` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))
4		fveq2 4681	. . . . . . . . . . . . . 14 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (BaseSet` U) = (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))
5		vacn.11	. . . . . . . . . . . . . 14 |- X = (BaseSet` U)
6	4, 5	syl5eq 1940	. . . . . . . . . . . . 13 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> X = (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))
7		xpeq1 4016	. . . . . . . . . . . . 13 |- (X = (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) -> (X X. X) = ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. X))
8	6, 7	syl 12	. . . . . . . . . . . 12 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (X X. X) = ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. X))
9		xpeq2 4017	. . . . . . . . . . . . 13 |- (X = (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) -> ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. X) = ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))))
10	6, 9	syl 12	. . . . . . . . . . . 12 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. X) = ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))))
11	8, 10	eqtrd 1925	. . . . . . . . . . 11 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (X X. X) = ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))))
12	11	eleq2d 1964	. . . . . . . . . 10 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (x e. (X X. X) <-> x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))))
13	11	eleq2d 1964	. . . . . . . . . 10 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (y e. (X X. X) <-> y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))))
14	12, 13	anbi12d 690	. . . . . . . . 9 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> ((x e. (X X. X) /\ y e. (X X. X)) <-> (x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))))))
15		fveq2 4681	. . . . . . . . . . . . . . 15 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (IndMet` U) = (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))
16		vacn.13	. . . . . . . . . . . . . . 15 |- C = (IndMet` U)
17	15, 16	syl5eq 1940	. . . . . . . . . . . . . 14 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> C = (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))
18	17	opreqd 4899	. . . . . . . . . . . . 13 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> ((1st` x)C(1st` y)) = ((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)))
19		preq1 3098	. . . . . . . . . . . . 13 |- (((1st` x)C(1st` y)) = ((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)) -> {((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))} = {((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)C(2nd` y))})
20	18, 19	syl 12	. . . . . . . . . . . 12 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> {((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))} = {((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)C(2nd` y))})
21	17	opreqd 4899	. . . . . . . . . . . . 13 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> ((2nd` x)C(2nd` y)) = ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y)))
22		preq2 3099	. . . . . . . . . . . . 13 |- (((2nd` x)C(2nd` y)) = ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y)) -> {((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)C(2nd` y))} = {((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))})
23	21, 22	syl 12	. . . . . . . . . . . 12 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> {((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)C(2nd` y))} = {((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))})
24	20, 23	eqtrd 1925	. . . . . . . . . . 11 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> {((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))} = {((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))})
25		supeq1 5665	. . . . . . . . . . 11 |- ({((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))} = {((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))} -> sup({((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))
26	24, 25	syl 12	. . . . . . . . . 10 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> sup({((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))
27	26	eqeq2d 1895	. . . . . . . . 9 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (z = sup({((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) <-> z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < )))
28	14, 27	anbi12d 690	. . . . . . . 8 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (((x e. (X X. X) /\ y e. (X X. X)) /\ z = sup({((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))}, RR, < )) <-> ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))))
29	28	oprabbidv 4922	. . . . . . 7 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> {<.<.x, y>., z>. | ((x e. (X X. X) /\ y e. (X X. X)) /\ z = sup({((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))} = {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))})
30		vacn.14	. . . . . . 7 |- D = {<.<.x, y>., z>. | ((x e. (X X. X) /\ y e. (X X. X)) /\ z = sup({((1st` x)C(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
31	29, 30	syl5eq 1940	. . . . . 6 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> D = {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))})
32	31	fveq2d 4685	. . . . 5 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (Open` D) = (Open` {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))}))
33		vacn.16	. . . . 5 |- K = (Open` D)
34	32, 33	syl5eq 1940	. . . 4 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> K = (Open` {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))}))
35	17	fveq2d 4685	. . . . 5 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (Open` C) = (Open` (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))))
36		vacn.15	. . . . 5 |- J = (Open` C)
37	35, 36	syl5eq 1940	. . . 4 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> J = (Open` (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))))
38	34, 37	opreq12d 4900	. . 3 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (K Cn J) = ((Open` {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))}) Cn (Open` (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))))
39	3, 38	eleq12d 1965	. 2 |- (U = if(U e. NrmCVec, U, <.<. + , x. >., abs>.) -> (G e. (K Cn J) <-> (+v` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) e. ((Open` {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))}) Cn (Open` (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))))))
40		eqid 1884	. . 3 |- (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) = (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))
41		eqid 1884	. . 3 |- (norm` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) = (norm` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))
42		eqid 1884	. . 3 |- (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) = (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))
43		elimnvu 9647	. . 3 |- if(U e. NrmCVec, U, <.<. + , x. >., abs>.) e. NrmCVec
44		eqid 1884	. . 3 |- (+v` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) = (+v` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))
45		eqid 1884	. . 3 |- {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))} = {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))}
46		eqid 1884	. . 3 |- (Open` (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) = (Open` (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))
47		eqid 1884	. . 3 |- (Open` {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))}) = (Open` {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))})
48		eqid 1884	. . 3 |- {<.k, r>. | (k e. NN /\ r = (1st` (h` k)))} = {<.k, r>. | (k e. NN /\ r = (1st` (h` k)))}
49		eqid 1884	. . 3 |- {<.k, r>. | (k e. NN /\ r = (2nd` (h` k)))} = {<.k, r>. | (k e. NN /\ r = (2nd` (h` k)))}
50		eqid 1884	. . 3 |- {<.k, r>. | (k e. NN /\ r = ((+v` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))` (h` k)))} = {<.k, r>. | (k e. NN /\ r = ((+v` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))` (h` k)))}
51	40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50	vacnlem6 9672	. 2 |- (+v` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) e. ((Open` {<.<.x, y>., z>. | ((x e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))) /\ y e. ((BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)) X. (BaseSet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.)))) /\ z = sup({((1st` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(1st` y)), ((2nd` x)(IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))(2nd` y))}, RR, < ))}) Cn (Open` (IndMet` if(U e. NrmCVec, U, <.<. + , x. >., abs>.))))
52	39, 51	dedth 3011	1 |- (U e. NrmCVec -> G e. (K Cn J))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  ifcif 2982  {cpr 3045  <.cop 3046  {copab 3395   X. cxp 3984  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  supcsup 5663  RRcr 6385   + caddc 6389   x. cmul 6391  NNcn 6449   < clt 6653  abscabs 8000   Cn ccn 9028  Opencopn 9069  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  normcnm 9541  IndMetcims 9542

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-reg 5695  ax-inf2 5731  ax-ac 5906

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-iin 3258  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-r1 5750  df-rank 5751  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-fl 7463  df-uz 7587  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-lm 9200  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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