Theorem heiborlem42 15996

Description: Lemma for heibor 15997. Use the deduction theorem. Since every open cover has a finite subcover, J is compact.

Hypotheses

Ref	Expression
heibor.1	|- J = (Open` M)
heibor.2	|- X = dom dom M

Assertion

Ref	Expression
heiborlem42	|- ((M e. CMet /\ M e. TotBnd) -> J e. Comp)

Proof of Theorem heiborlem42

Step	Hyp	Ref	Expression
1		iscomp 10330	. 2 |- (J e. Comp <-> (J e. Top /\ A.u e. ~P J(U.J = U.u -> E.w e. (~Pu i^i Fin)U.J = U.w)))
2		cmsmet 9239	. . . 4 |- (M e. CMet -> M e. Met)
3		heibor.1	. . . . 5 |- J = (Open` M)
4	3	opntop 9147	. . . 4 |- (M e. Met -> J e. Top)
5	2, 4	syl 12	. . 3 |- (M e. CMet -> J e. Top)
6	5	adantr 425	. 2 |- ((M e. CMet /\ M e. TotBnd) -> J e. Top)
7		dmeq 4157	. . . . . . . . . . 11 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> dom M = dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))
8	7	dmeqd 4159	. . . . . . . . . 10 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> dom dom M = dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))
9	8	eqeq1d 1892	. . . . . . . . 9 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (dom dom M = U.w <-> dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.w))
10	9	rexbidv 2124	. . . . . . . 8 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (E.w e. (~Pu i^i Fin)dom dom M = U.w <-> E.w e. (~Pu i^i Fin)dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.w))
11		pweq 3036	. . . . . . . . . 10 |- (u = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> ~Pu = ~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}))
12	11	ineq1d 2795	. . . . . . . . 9 |- (u = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> (~Pu i^i Fin) = (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin))
13	12	rexeqdv 2270	. . . . . . . 8 |- (u = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> (E.w e. (~Pu i^i Fin)dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.w <-> E.w e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.w))
14		eqid 1884	. . . . . . . . 9 |- (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) = (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))
15		eqid 1884	. . . . . . . . 9 |- dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))
16		eleq1 1957	. . . . . . . . . . . . . 14 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (M e. CMet <-> if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet))
17		eleq1 1957	. . . . . . . . . . . . . 14 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (M e. TotBnd <-> if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd))
18	16, 17	anbi12d 690	. . . . . . . . . . . . 13 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> ((M e. CMet /\ M e. TotBnd) <-> (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd)))
19		fveq2 4681	. . . . . . . . . . . . . . 15 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (Open` M) = (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))))
20	19	sseq2d 2645	. . . . . . . . . . . . . 14 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (u C_ (Open` M) <-> u C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))))
21	8	eqeq1d 1892	. . . . . . . . . . . . . 14 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (dom dom M = U.u <-> dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.u))
22	20, 21	anbi12d 690	. . . . . . . . . . . . 13 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> ((u C_ (Open` M) /\ dom dom M = U.u) <-> (u C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.u)))
23	18, 22	anbi12d 690	. . . . . . . . . . . 12 |- (M = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)) <-> ((if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd) /\ (u C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.u))))
24		sseq1 2637	. . . . . . . . . . . . . 14 |- (u = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> (u C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) <-> if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))))
25		unieq 3185	. . . . . . . . . . . . . . 15 |- (u = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> U.u = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}))
26	25	eqeq2d 1895	. . . . . . . . . . . . . 14 |- (u = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> (dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.u <-> dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)})))
27	24, 26	anbi12d 690	. . . . . . . . . . . . 13 |- (u = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> ((u C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.u) <-> (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}))))
28	27	anbi2d 678	. . . . . . . . . . . 12 |- (u = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> (((if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd) /\ (u C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.u)) <-> ((if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd) /\ (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)})))))
29		eleq1 1957	. . . . . . . . . . . . . 14 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> ((/) e. CMet <-> if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet))
30		eleq1 1957	. . . . . . . . . . . . . 14 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> ((/) e. TotBnd <-> if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd))
31	29, 30	anbi12d 690	. . . . . . . . . . . . 13 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (((/) e. CMet /\ (/) e. TotBnd) <-> (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd)))
32		fveq2 4681	. . . . . . . . . . . . . . 15 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (Open` (/)) = (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))))
33	32	sseq2d 2645	. . . . . . . . . . . . . 14 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> ({(/)} C_ (Open` (/)) <-> {(/)} C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))))
34		dmeq 4157	. . . . . . . . . . . . . . . 16 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> dom (/) = dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))
35	34	dmeqd 4159	. . . . . . . . . . . . . . 15 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> dom dom (/) = dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))
36	35	eqeq1d 1892	. . . . . . . . . . . . . 14 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (dom dom (/) = U.{(/)} <-> dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.{(/)}))
37	33, 36	anbi12d 690	. . . . . . . . . . . . 13 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> (({(/)} C_ (Open` (/)) /\ dom dom (/) = U.{(/)}) <-> ({(/)} C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.{(/)})))
38	31, 37	anbi12d 690	. . . . . . . . . . . 12 |- ((/) = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) -> ((((/) e. CMet /\ (/) e. TotBnd) /\ ({(/)} C_ (Open` (/)) /\ dom dom (/) = U.{(/)})) <-> ((if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd) /\ ({(/)} C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.{(/)}))))
39		sseq1 2637	. . . . . . . . . . . . . 14 |- ({(/)} = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> ({(/)} C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) <-> if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))))
40		unieq 3185	. . . . . . . . . . . . . . 15 |- ({(/)} = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> U.{(/)} = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}))
41	40	eqeq2d 1895	. . . . . . . . . . . . . 14 |- ({(/)} = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> (dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.{(/)} <-> dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)})))
42	39, 41	anbi12d 690	. . . . . . . . . . . . 13 |- ({(/)} = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> (({(/)} C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.{(/)}) <-> (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}))))
43	42	anbi2d 678	. . . . . . . . . . . 12 |- ({(/)} = if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) -> (((if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd) /\ ({(/)} C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.{(/)})) <-> ((if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd) /\ (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)})))))
44		dm0 4170	. . . . . . . . . . . . . . . . 17 |- dom (/) = (/)
45	44	dmeqi 4158	. . . . . . . . . . . . . . . 16 |- dom dom (/) = dom (/)
46	45, 44	eqtr2i 1909	. . . . . . . . . . . . . . 15 |- (/) = dom dom (/)
47		0met 9102	. . . . . . . . . . . . . . 15 |- (/) e. Met
48		f00 4601	. . . . . . . . . . . . . . . . . 18 |- (g:NN-->(/) <-> (g = (/) /\ NN = (/)))
49	48	simprbi 353	. . . . . . . . . . . . . . . . 17 |- (g:NN-->(/) -> NN = (/))
50		1nn 7117	. . . . . . . . . . . . . . . . . . 19 |- 1 e. NN
51		n0i 2880	. . . . . . . . . . . . . . . . . . 19 |- (1 e. NN -> -. NN = (/))
52	50, 51	ax-mp 7	. . . . . . . . . . . . . . . . . 18 |- -. NN = (/)
53	52	pm2.21i 93	. . . . . . . . . . . . . . . . 17 |- (NN = (/) -> E.z e. (/) g(~~>m` (/))z)
54	49, 53	syl 12	. . . . . . . . . . . . . . . 16 |- (g:NN-->(/) -> E.z e. (/) g(~~>m` (/))z)
55	54	adantl 424	. . . . . . . . . . . . . . 15 |- ((g e. (Cau` (/)) /\ g:NN-->(/)) -> E.z e. (/) g(~~>m` (/))z)
56	46, 47, 55	iscms2i 9273	. . . . . . . . . . . . . 14 |- (/) e. CMet
57	46	istotbnd2 15934	. . . . . . . . . . . . . . . 16 |- ((/) e. Met -> ((/) e. TotBnd <-> A.d e. RR+ E.w e. Fin (U.w = (/) /\ A.y e. w E.z e. (/) y = (z( ball ` (/))d))))
58	47, 57	ax-mp 7	. . . . . . . . . . . . . . 15 |- ((/) e. TotBnd <-> A.d e. RR+ E.w e. Fin (U.w = (/) /\ A.y e. w E.z e. (/) y = (z( ball ` (/))d)))
59		emfin 10165	. . . . . . . . . . . . . . . . 17 |- (/) e. Fin
60		uni0 3205	. . . . . . . . . . . . . . . . . 18 |- U.(/) = (/)
61		ral0 2974	. . . . . . . . . . . . . . . . . 18 |- A.y e. (/) E.z e. (/) y = (z( ball ` (/))d)
62	60, 61	pm3.2i 307	. . . . . . . . . . . . . . . . 17 |- (U.(/) = (/) /\ A.y e. (/) E.z e. (/) y = (z( ball ` (/))d))
63		unieq 3185	. . . . . . . . . . . . . . . . . . . 20 |- (w = (/) -> U.w = U.(/))
64	63	eqeq1d 1892	. . . . . . . . . . . . . . . . . . 19 |- (w = (/) -> (U.w = (/) <-> U.(/) = (/)))
65		raleq 2266	. . . . . . . . . . . . . . . . . . 19 |- (w = (/) -> (A.y e. w E.z e. (/) y = (z( ball ` (/))d) <-> A.y e. (/) E.z e. (/) y = (z( ball ` (/))d)))
66	64, 65	anbi12d 690	. . . . . . . . . . . . . . . . . 18 |- (w = (/) -> ((U.w = (/) /\ A.y e. w E.z e. (/) y = (z( ball ` (/))d)) <-> (U.(/) = (/) /\ A.y e. (/) E.z e. (/) y = (z( ball ` (/))d))))
67	66	rcla4ev 2381	. . . . . . . . . . . . . . . . 17 |- (((/) e. Fin /\ (U.(/) = (/) /\ A.y e. (/) E.z e. (/) y = (z( ball ` (/))d))) -> E.w e. Fin (U.w = (/) /\ A.y e. w E.z e. (/) y = (z( ball ` (/))d)))
68	59, 62, 67	mp2an 761	. . . . . . . . . . . . . . . 16 |- E.w e. Fin (U.w = (/) /\ A.y e. w E.z e. (/) y = (z( ball ` (/))d))
69	68	a1i 8	. . . . . . . . . . . . . . 15 |- (d e. RR+ -> E.w e. Fin (U.w = (/) /\ A.y e. w E.z e. (/) y = (z( ball ` (/))d)))
70	58, 69	mprgbir 2163	. . . . . . . . . . . . . 14 |- (/) e. TotBnd
71	56, 70	pm3.2i 307	. . . . . . . . . . . . 13 |- ((/) e. CMet /\ (/) e. TotBnd)
72		eqid 1884	. . . . . . . . . . . . . . . . 17 |- (Open` (/)) = (Open` (/))
73	72	opn0 9150	. . . . . . . . . . . . . . . 16 |- ((/) e. Met -> (/) e. (Open` (/)))
74	47, 73	ax-mp 7	. . . . . . . . . . . . . . 15 |- (/) e. (Open` (/))
75		snssi 3129	. . . . . . . . . . . . . . 15 |- ((/) e. (Open` (/)) -> {(/)} C_ (Open` (/)))
76	74, 75	ax-mp 7	. . . . . . . . . . . . . 14 |- {(/)} C_ (Open` (/))
77		0ex 3446	. . . . . . . . . . . . . . . 16 |- (/) e. _V
78	77	unisn 3193	. . . . . . . . . . . . . . 15 |- U.{(/)} = (/)
79	44, 45, 78	3eqtr4i 1921	. . . . . . . . . . . . . 14 |- dom dom (/) = U.{(/)}
80	76, 79	pm3.2i 307	. . . . . . . . . . . . 13 |- ({(/)} C_ (Open` (/)) /\ dom dom (/) = U.{(/)})
81	71, 80	pm3.2i 307	. . . . . . . . . . . 12 |- (((/) e. CMet /\ (/) e. TotBnd) /\ ({(/)} C_ (Open` (/)) /\ dom dom (/) = U.{(/)}))
82	23, 28, 38, 43, 81	elimhyp2v 3022	. . . . . . . . . . 11 |- ((if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd) /\ (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)})))
83	82	simpli 347	. . . . . . . . . 10 |- (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet /\ if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd)
84	83	simpli 347	. . . . . . . . 9 |- if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. CMet
85	83	simpri 351	. . . . . . . . 9 |- if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) e. TotBnd
86	82	simpri 351	. . . . . . . . . 10 |- (if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/))) /\ dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}))
87	86	simpli 347	. . . . . . . . 9 |- if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) C_ (Open` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))
88	86	simpri 351	. . . . . . . . 9 |- dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)})
89		eqid 1884	. . . . . . . . 9 |- {s e. ~Pdom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) | -. E.v e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)s C_ U.v} = {s e. ~Pdom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) | -. E.v e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)s C_ U.v}
90		eqid 1884	. . . . . . . . 9 |- {<.x, r>. | ((x e. dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) /\ r e. RR+) /\ (x( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))r) e. {s e. ~Pdom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) | -. E.v e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)s C_ U.v})} = {<.x, r>. | ((x e. dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) /\ r e. RR+) /\ (x( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))r) e. {s e. ~Pdom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) | -. E.v e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)s C_ U.v})}
91		eqid 1884	. . . . . . . . 9 |- {<.<.n, a>., h>. | ((n e. NN /\ a e. {<.x, r>. | ((x e. dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) /\ r e. RR+) /\ (x( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))r) e. {s e. ~Pdom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) | -. E.v e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)s C_ U.v})}) /\ h = {<.x, r>. | ((x e. dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) /\ r = (1 / (2^n))) /\ ((x( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))r) e. {s e. ~Pdom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) | -. E.v e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)s C_ U.v} /\ ((x( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))r) i^i (( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))` a)) =/= (/)))})} = {<.<.n, a>., h>. | ((n e. NN /\ a e. {<.x, r>. | ((x e. dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) /\ r e. RR+) /\ (x( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))r) e. {s e. ~Pdom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) | -. E.v e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)s C_ U.v})}) /\ h = {<.x, r>. | ((x e. dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) /\ r = (1 / (2^n))) /\ ((x( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))r) e. {s e. ~Pdom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) | -. E.v e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)s C_ U.v} /\ ((x( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))r) i^i (( ball ` if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)))` a)) =/= (/)))})}
92	14, 15, 84, 85, 87, 88, 89, 90, 91	heiborlem41 15995	. . . . . . . 8 |- E.w e. (~Pif(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), u, {(/)}) i^i Fin)dom dom if(((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)), M, (/)) = U.w
93	10, 13, 92	dedth2v 3018	. . . . . . 7 |- (((M e. CMet /\ M e. TotBnd) /\ (u C_ (Open` M) /\ dom dom M = U.u)) -> E.w e. (~Pu i^i Fin)dom dom M = U.w)
94	3	sseq2i 2642	. . . . . . . 8 |- (u C_ J <-> u C_ (Open` M))
95	94	biimpi 168	. . . . . . 7 |- (u C_ J -> u C_ (Open` M))
96	93, 95	sylanr1 511	. . . . . 6 |- (((M e. CMet /\ M e. TotBnd) /\ (u C_ J /\ dom dom M = U.u)) -> E.w e. (~Pu i^i Fin)dom dom M = U.w)
97	96	expr 418	. . . . 5 |- (((M e. CMet /\ M e. TotBnd) /\ u C_ J) -> (dom dom M = U.u -> E.w e. (~Pu i^i Fin)dom dom M = U.w))
98		elpwi 3039	. . . . 5 |- (u e. ~PJ -> u C_ J)
99	97, 98	sylan2 500	. . . 4 |- (((M e. CMet /\ M e. TotBnd) /\ u e. ~PJ) -> (dom dom M = U.u -> E.w e. (~Pu i^i Fin)dom dom M = U.w))
100		eqid 1884	. . . . . . . 8 |- dom dom M = dom dom M
101	100, 3	uniopn2 9138	. . . . . . 7 |- (M e. Met -> U.J = dom dom M)
102	2, 101	syl 12	. . . . . 6 |- (M e. CMet -> U.J = dom dom M)
103	102	eqeq1d 1892	. . . . 5 |- (M e. CMet -> (U.J = U.u <-> dom dom M = U.u))
104	103	ad2antrr 440	. . . 4 |- (((M e. CMet /\ M e. TotBnd) /\ u e. ~PJ) -> (U.J = U.u <-> dom dom M = U.u))
105	102	eqeq1d 1892	. . . . . 6 |- (M e. CMet -> (U.J = U.w <-> dom dom M = U.w))
106	105	rexbidv 2124	. . . . 5 |- (M e. CMet -> (E.w e. (~Pu i^i Fin)U.J = U.w <-> E.w e. (~Pu i^i Fin)dom dom M = U.w))
107	106	ad2antrr 440	. . . 4 |- (((M e. CMet /\ M e. TotBnd) /\ u e. ~PJ) -> (E.w e. (~Pu i^i Fin)U.J = U.w <-> E.w e. (~Pu i^i Fin)dom dom M = U.w))
108	99, 104, 107	3imtr4d 602	. . 3 |- (((M e. CMet /\ M e. TotBnd) /\ u e. ~PJ) -> (U.J = U.u -> E.w e. (~Pu i^i Fin)U.J = U.w))
109	108	r19.21aiva 2176	. 2 |- ((M e. CMet /\ M e. TotBnd) -> A.u e. ~P J(U.J = U.u -> E.w e. (~Pu i^i Fin)U.J = U.w))
110	1, 6, 109	sylanbrc 527	1 |- ((M e. CMet /\ M e. TotBnd) -> J e. Comp)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   i^i cin 2592   C_ wss 2593  (/)c0 2875  ifcif 2982  ~Pcpw 3032  {csn 3044  U.cuni 3177   class class class wbr 3338  {copab 3395  dom cdm 3986  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  Fincfn 5426  1c1 6387   / cdiv 6447  NNcn 6449  RR+crp 6453  2c2 7145  ^cexp 7811  Topctop 8857  Metcme 9066   ball cbl 9068  Opencopn 9069  ~~>mclm 9197  Caucca 9198  CMetcms 9199  Compccomp 10328  TotBndctotbnd 15930

This theorem is referenced by:  heibor 15997

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  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-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-iso 4015  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-fin 5430  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-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-top 8861  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-cau 9201  df-cmet 9202  df-comp 10329  df-totbnd 15932  df-bnd 15938

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