Theorem bcth 9310

Description: Baire's Category Theorem. If a nonempty metric space is complete, it is nonmeager in itself. In other words, the metric space cannot be the countable union of rare closed subsets (where rare means having an empty interior), so some subset M` k must have a nonempty interior. Theorem 4.7-2 of [Kreyszig] p. 247. (The terminology "meager" and "nonmeager" is used by Kreyszig to replace Baire's "of the first category" and "of the second category." The latter terms are going out of favor to avoid confusion with category theory.)

Hypotheses

Ref	Expression
bcth.1	|- X = dom dom D
bcth.2	|- J = (Open` D)

Assertion

Ref	Expression
bcth	|- (((D e. CMet /\ X =/= (/) /\ M:NN-->~PX) /\ (U.ran M = X /\ ran M C_ (Clsd` J))) -> E.k e. NN ((int` J)` (M` k)) =/= (/))

Distinct variable groups:   D,k   k,J   k,M   k,X

Proof of Theorem bcth

Step	Hyp	Ref	Expression
1		dmeq 4157	. . . . . . . . 9 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> dom D = dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
2	1	dmeqd 4159	. . . . . . . 8 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> dom dom D = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
3		bcth.1	. . . . . . . 8 |- X = dom dom D
4	2, 3	syl5eq 1940	. . . . . . 7 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> X = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
5		pweq 3036	. . . . . . 7 |- (X = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ~PX = ~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
6	4, 5	syl 12	. . . . . 6 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ~PX = ~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
7		feq3 4553	. . . . . 6 |- (~PX = ~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (M:NN-->~PX <-> M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))
8	6, 7	syl 12	. . . . 5 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (M:NN-->~PX <-> M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))
9	4	eqeq2d 1895	. . . . . . 7 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (U.ran M = X <-> U.ran M = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))
10		fveq2 4681	. . . . . . . . . 10 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (Open` D) = (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))
11		bcth.2	. . . . . . . . . 10 |- J = (Open` D)
12	10, 11	syl5eq 1940	. . . . . . . . 9 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> J = (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))
13	12	fveq2d 4685	. . . . . . . 8 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (Clsd` J) = (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - )))))
14	13	sseq2d 2645	. . . . . . 7 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (ran M C_ (Clsd` J) <-> ran M C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))))
15	9, 14	anbi12d 690	. . . . . 6 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ((U.ran M = X /\ ran M C_ (Clsd` J)) <-> (U.ran M = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran M C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - )))))))
16	12	fveq2d 4685	. . . . . . . . 9 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (int` J) = (int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - )))))
17	16	fveq1d 4683	. . . . . . . 8 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ((int` J)` (M` k)) = ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)))
18	17	neeq1d 2028	. . . . . . 7 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (((int` J)` (M` k)) =/= (/) <-> ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) =/= (/)))
19	18	rexbidv 2124	. . . . . 6 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (E.k e. NN ((int` J)` (M` k)) =/= (/) <-> E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) =/= (/)))
20	15, 19	imbi12d 688	. . . . 5 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (((U.ran M = X /\ ran M C_ (Clsd` J)) -> E.k e. NN ((int` J)` (M` k)) =/= (/)) <-> ((U.ran M = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran M C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))) -> E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) =/= (/))))
21	8, 20	imbi12d 688	. . . 4 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ((M:NN-->~PX -> ((U.ran M = X /\ ran M C_ (Clsd` J)) -> E.k e. NN ((int` J)` (M` k)) =/= (/))) <-> (M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ((U.ran M = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran M C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))) -> E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) =/= (/)))))
22		rneq 4186	. . . . . . . . 9 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> ran M = ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})))
23	22	unieqd 3188	. . . . . . . 8 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> U.ran M = U.ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})))
24	23	eqeq1d 1892	. . . . . . 7 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> (U.ran M = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) <-> U.ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))
25	22	sseq1d 2644	. . . . . . 7 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> (ran M C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - )))) <-> ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))))
26	24, 25	anbi12d 690	. . . . . 6 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> ((U.ran M = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran M C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))) <-> (U.ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - )))))))
27		fveq1 4680	. . . . . . . . 9 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> (M` k) = (if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)}))` k))
28	27	fveq2d 4685	. . . . . . . 8 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) = ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)}))` k)))
29	28	neeq1d 2028	. . . . . . 7 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> (((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) =/= (/) <-> ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)}))` k)) =/= (/)))
30	29	rexbidv 2124	. . . . . 6 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> (E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) =/= (/) <-> E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)}))` k)) =/= (/)))
31	26, 30	imbi12d 688	. . . . 5 |- (M = if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) -> (((U.ran M = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran M C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))) -> E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) =/= (/)) <-> ((U.ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))) -> E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)}))` k)) =/= (/))))
32		eleq1 1957	. . . . . . . . 9 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (D e. CMet <-> if((D e. CMet /\ X =/= (/)), D, (abs o. - )) e. CMet))
33	4	neeq1d 2028	. . . . . . . . 9 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (X =/= (/) <-> dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) =/= (/)))
34	32, 33	anbi12d 690	. . . . . . . 8 |- (D = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ((D e. CMet /\ X =/= (/)) <-> (if((D e. CMet /\ X =/= (/)), D, (abs o. - )) e. CMet /\ dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) =/= (/))))
35		eleq1 1957	. . . . . . . . 9 |- ((abs o. - ) = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ((abs o. - ) e. CMet <-> if((D e. CMet /\ X =/= (/)), D, (abs o. - )) e. CMet))
36		dmeq 4157	. . . . . . . . . . 11 |- ((abs o. - ) = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> dom (abs o. - ) = dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
37	36	dmeqd 4159	. . . . . . . . . 10 |- ((abs o. - ) = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> dom dom (abs o. - ) = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
38	37	neeq1d 2028	. . . . . . . . 9 |- ((abs o. - ) = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (dom dom (abs o. - ) =/= (/) <-> dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) =/= (/)))
39	35, 38	anbi12d 690	. . . . . . . 8 |- ((abs o. - ) = if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> (((abs o. - ) e. CMet /\ dom dom (abs o. - ) =/= (/)) <-> (if((D e. CMet /\ X =/= (/)), D, (abs o. - )) e. CMet /\ dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) =/= (/))))
40		eqid 1884	. . . . . . . . . 10 |- (abs o. - ) = (abs o. - )
41	40	cncms 9276	. . . . . . . . 9 |- (abs o. - ) e. CMet
42		0cn 6481	. . . . . . . . . . 11 |- 0 e. CC
43		ne0i 2881	. . . . . . . . . . 11 |- (0 e. CC -> CC =/= (/))
44	42, 43	ax-mp 7	. . . . . . . . . 10 |- CC =/= (/)
45	40	cnmetba 9181	. . . . . . . . . . 11 |- CC = dom dom (abs o. - )
46	45	neeq1i 2026	. . . . . . . . . 10 |- (CC =/= (/) <-> dom dom (abs o. - ) =/= (/))
47	44, 46	mpbi 206	. . . . . . . . 9 |- dom dom (abs o. - ) =/= (/)
48	41, 47	pm3.2i 307	. . . . . . . 8 |- ((abs o. - ) e. CMet /\ dom dom (abs o. - ) =/= (/))
49	34, 39, 48	elimhyp 3021	. . . . . . 7 |- (if((D e. CMet /\ X =/= (/)), D, (abs o. - )) e. CMet /\ dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) =/= (/))
50	49	simpli 347	. . . . . 6 |- if((D e. CMet /\ X =/= (/)), D, (abs o. - )) e. CMet
51	49	simpri 351	. . . . . 6 |- dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) =/= (/)
52		eqid 1884	. . . . . 6 |- dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))
53		eqid 1884	. . . . . 6 |- (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))) = (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
54		0ex 3446	. . . . . . . . 9 |- (/) e. _V
55	54	fconst 4602	. . . . . . . 8 |- (NN X. {(/)}):NN-->{(/)}
56		0elpw 3473	. . . . . . . . 9 |- (/) e. ~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))
57		snssi 3129	. . . . . . . . 9 |- ((/) e. ~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> {(/)} C_ ~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
58	56, 57	ax-mp 7	. . . . . . . 8 |- {(/)} C_ ~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))
59		fss 4571	. . . . . . . 8 |- (((NN X. {(/)}):NN-->{(/)} /\ {(/)} C_ ~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))) -> (NN X. {(/)}):NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )))
60	55, 58, 59	mp2an 761	. . . . . . 7 |- (NN X. {(/)}):NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))
61	60	elimf 4561	. . . . . 6 |- if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})):NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - ))
62	50, 51, 52, 53, 61	bcthlem33 9309	. . . . 5 |- ((U.ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)})) C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))) -> E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (if(M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )), M, (NN X. {(/)}))` k)) =/= (/))
63	31, 62	dedth 3011	. . . 4 |- (M:NN-->~Pdom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) -> ((U.ran M = dom dom if((D e. CMet /\ X =/= (/)), D, (abs o. - )) /\ ran M C_ (Clsd` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))) -> E.k e. NN ((int` (Open` if((D e. CMet /\ X =/= (/)), D, (abs o. - ))))` (M` k)) =/= (/)))
64	21, 63	dedth 3011	. . 3 |- ((D e. CMet /\ X =/= (/)) -> (M:NN-->~PX -> ((U.ran M = X /\ ran M C_ (Clsd` J)) -> E.k e. NN ((int` J)` (M` k)) =/= (/))))
65	64	3impia 1064	. 2 |- ((D e. CMet /\ X =/= (/) /\ M:NN-->~PX) -> ((U.ran M = X /\ ran M C_ (Clsd` J)) -> E.k e. NN ((int` J)` (M` k)) =/= (/)))
66	65	imp 377	1 |- (((D e. CMet /\ X =/= (/) /\ M:NN-->~PX) /\ (U.ran M = X /\ ran M C_ (Clsd` J))) -> E.k e. NN ((int` J)` (M` k)) =/= (/))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   C_ wss 2593  (/)c0 2875  ifcif 2982  ~Pcpw 3032  {csn 3044  U.cuni 3177   X. cxp 3984  dom cdm 3986  ran crn 3987   o. ccom 3990  -->wf 3994  ` cfv 3998  CCcc 6384  0cc0 6386   - cmin 6445  NNcn 6449  abscabs 8000  Clsdccld 8936  intcnt 8937  Opencopn 9069  CMetcms 9199

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  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-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-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-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-clim 8235  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-cau 9201  df-cmet 9202

Copyright terms: Public domain