Theorem climsupi 8415

Description: A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180.

Hypotheses

Ref	Expression
climsup.1	|- F:NN-->RR
climsup.2	|- (k e. NN -> (F` k) <_ (F` (k + 1)))
climsup.3	|- E.x e. RR A.k e. NN (F` k) <_ x

Assertion

Ref	Expression
climsupi	|- F ~~> sup(ran F, RR, < )

Distinct variable group:   x,k,F

Proof of Theorem climsupi

Step	Hyp	Ref	Expression
1		climsup.1	. . . 4 |- F:NN-->RR
2		axresscn 6420	. . . 4 |- RR C_ CC
3		fss 4571	. . . 4 |- ((F:NN-->RR /\ RR C_ CC) -> F:NN-->CC)
4	1, 2, 3	mp2an 761	. . 3 |- F:NN-->CC
5		frn 4569	. . . . . . 7 |- (F:NN-->RR -> ran F C_ RR)
6	1, 5	ax-mp 7	. . . . . 6 |- ran F C_ RR
7		1nn 7117	. . . . . . . . 9 |- 1 e. NN
8		ne0i 2881	. . . . . . . . 9 |- (1 e. NN -> NN =/= (/))
9	7, 8	ax-mp 7	. . . . . . . 8 |- NN =/= (/)
10	1	fdmi 4568	. . . . . . . . 9 |- dom F = NN
11	10	neeq1i 2026	. . . . . . . 8 |- (dom F =/= (/) <-> NN =/= (/))
12	9, 11	mpbir 207	. . . . . . 7 |- dom F =/= (/)
13		dm0rn0 4175	. . . . . . . 8 |- (dom F = (/) <-> ran F = (/))
14	13	necon3bii 2032	. . . . . . 7 |- (dom F =/= (/) <-> ran F =/= (/))
15	12, 14	mpbi 206	. . . . . 6 |- ran F =/= (/)
16		climsup.3	. . . . . . 7 |- E.x e. RR A.k e. NN (F` k) <_ x
17		hbra1 2147	. . . . . . . . . . 11 |- (A.k e. NN (F` k) <_ x -> A.kA.k e. NN (F` k) <_ x)
18		ax-17 1317	. . . . . . . . . . 11 |- (y <_ x -> A.k y <_ x)
19		ra4 2155	. . . . . . . . . . . 12 |- (A.k e. NN (F` k) <_ x -> (k e. NN -> (F` k) <_ x))
20		breq1 3341	. . . . . . . . . . . . 13 |- ((F` k) = y -> ((F` k) <_ x <-> y <_ x))
21	20	biimpcd 172	. . . . . . . . . . . 12 |- ((F` k) <_ x -> ((F` k) = y -> y <_ x))
22	19, 21	syl6 25	. . . . . . . . . . 11 |- (A.k e. NN (F` k) <_ x -> (k e. NN -> ((F` k) = y -> y <_ x)))
23	17, 18, 22	r19.23ad 2213	. . . . . . . . . 10 |- (A.k e. NN (F` k) <_ x -> (E.k e. NN (F` k) = y -> y <_ x))
24		ffn 4562	. . . . . . . . . . . 12 |- (F:NN-->RR -> F Fn NN)
25	1, 24	ax-mp 7	. . . . . . . . . . 11 |- F Fn NN
26		fvelrnb 4719	. . . . . . . . . . 11 |- (F Fn NN -> (y e. ran F <-> E.k e. NN (F` k) = y))
27	25, 26	ax-mp 7	. . . . . . . . . 10 |- (y e. ran F <-> E.k e. NN (F` k) = y)
28	23, 27	syl5ib 223	. . . . . . . . 9 |- (A.k e. NN (F` k) <_ x -> (y e. ran F -> y <_ x))
29	28	r19.21aiv 2175	. . . . . . . 8 |- (A.k e. NN (F` k) <_ x -> A.y e. ran F y <_ x)
30	29	reximi 2198	. . . . . . 7 |- (E.x e. RR A.k e. NN (F` k) <_ x -> E.x e. RR A.y e. ran F y <_ x)
31	16, 30	ax-mp 7	. . . . . 6 |- E.x e. RR A.y e. ran F y <_ x
32	6, 15, 31	3pm3.2i 1048	. . . . 5 |- (ran F C_ RR /\ ran F =/= (/) /\ E.x e. RR A.y e. ran F y <_ x)
33	32	suprclii 7270	. . . 4 |- sup(ran F, RR, < ) e. RR
34	33	recni 6467	. . 3 |- sup(ran F, RR, < ) e. CC
35		climfnn 8352	. . 3 |- ((F:NN-->CC /\ sup(ran F, RR, < ) e. CC) -> (F ~~> sup(ran F, RR, < ) <-> A.x e. RR (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x))))
36	4, 34, 35	mp2an 761	. 2 |- (F ~~> sup(ran F, RR, < ) <-> A.x e. RR (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x)))
37		ltsubpos 6841	. . . . 5 |- ((x e. RR /\ sup(ran F, RR, < ) e. RR) -> (0 < x <-> (sup(ran F, RR, < ) - x) < sup(ran F, RR, < )))
38	33, 37	mpan2 760	. . . 4 |- (x e. RR -> (0 < x <-> (sup(ran F, RR, < ) - x) < sup(ran F, RR, < )))
39		ax-17 1317	. . . . . . . . . . . . 13 |- (z <_ x -> A.k z <_ x)
40		breq1 3341	. . . . . . . . . . . . . . 15 |- ((F` k) = z -> ((F` k) <_ x <-> z <_ x))
41	40	biimpcd 172	. . . . . . . . . . . . . 14 |- ((F` k) <_ x -> ((F` k) = z -> z <_ x))
42	19, 41	syl6 25	. . . . . . . . . . . . 13 |- (A.k e. NN (F` k) <_ x -> (k e. NN -> ((F` k) = z -> z <_ x)))
43	17, 39, 42	r19.23ad 2213	. . . . . . . . . . . 12 |- (A.k e. NN (F` k) <_ x -> (E.k e. NN (F` k) = z -> z <_ x))
44		fvelrnb 4719	. . . . . . . . . . . . 13 |- (F Fn NN -> (z e. ran F <-> E.k e. NN (F` k) = z))
45	25, 44	ax-mp 7	. . . . . . . . . . . 12 |- (z e. ran F <-> E.k e. NN (F` k) = z)
46	43, 45	syl5ib 223	. . . . . . . . . . 11 |- (A.k e. NN (F` k) <_ x -> (z e. ran F -> z <_ x))
47	46	r19.21aiv 2175	. . . . . . . . . 10 |- (A.k e. NN (F` k) <_ x -> A.z e. ran F z <_ x)
48	47	reximi 2198	. . . . . . . . 9 |- (E.x e. RR A.k e. NN (F` k) <_ x -> E.x e. RR A.z e. ran F z <_ x)
49	16, 48	ax-mp 7	. . . . . . . 8 |- E.x e. RR A.z e. ran F z <_ x
50	6, 15, 49	3pm3.2i 1048	. . . . . . 7 |- (ran F C_ RR /\ ran F =/= (/) /\ E.x e. RR A.z e. ran F z <_ x)
51	50	suprlubii 7272	. . . . . 6 |- (((sup(ran F, RR, < ) - x) e. RR /\ (sup(ran F, RR, < ) - x) < sup(ran F, RR, < )) -> E.y e. ran F(sup(ran F, RR, < ) - x) < y)
52		resubcl 6601	. . . . . . 7 |- ((sup(ran F, RR, < ) e. RR /\ x e. RR) -> (sup(ran F, RR, < ) - x) e. RR)
53	33, 52	mpan 759	. . . . . 6 |- (x e. RR -> (sup(ran F, RR, < ) - x) e. RR)
54	51, 53	sylan 497	. . . . 5 |- ((x e. RR /\ (sup(ran F, RR, < ) - x) < sup(ran F, RR, < )) -> E.y e. ran F(sup(ran F, RR, < ) - x) < y)
55	54	ex 402	. . . 4 |- (x e. RR -> ((sup(ran F, RR, < ) - x) < sup(ran F, RR, < ) -> E.y e. ran F(sup(ran F, RR, < ) - x) < y))
56	38, 55	sylbid 220	. . 3 |- (x e. RR -> (0 < x -> E.y e. ran F(sup(ran F, RR, < ) - x) < y))
57	53	adantr 425	. . . . . . . . . . . . . 14 |- ((x e. RR /\ (sup(ran F, RR, < ) - x) < y) -> (sup(ran F, RR, < ) - x) e. RR)
58	57	ad2antrr 440	. . . . . . . . . . . . 13 |- ((((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) /\ (k e. NN /\ j <_ k)) -> (sup(ran F, RR, < ) - x) e. RR)
59	1	ffvelrni 4788	. . . . . . . . . . . . . . 15 |- (j e. NN -> (F` j) e. RR)
60	59	adantr 425	. . . . . . . . . . . . . 14 |- ((j e. NN /\ (F` j) = y) -> (F` j) e. RR)
61	60	ad2antlr 441	. . . . . . . . . . . . 13 |- ((((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) /\ (k e. NN /\ j <_ k)) -> (F` j) e. RR)
62	1	ffvelrni 4788	. . . . . . . . . . . . . 14 |- (k e. NN -> (F` k) e. RR)
63	62	ad2antrl 442	. . . . . . . . . . . . 13 |- ((((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) /\ (k e. NN /\ j <_ k)) -> (F` k) e. RR)
64		breq2 3342	. . . . . . . . . . . . . . . 16 |- ((F` j) = y -> ((sup(ran F, RR, < ) - x) < (F` j) <-> (sup(ran F, RR, < ) - x) < y))
65	64	biimparc 463	. . . . . . . . . . . . . . 15 |- (((sup(ran F, RR, < ) - x) < y /\ (F` j) = y) -> (sup(ran F, RR, < ) - x) < (F` j))
66	65	ad2ant2l 444	. . . . . . . . . . . . . 14 |- (((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) -> (sup(ran F, RR, < ) - x) < (F` j))
67	66	adantr 425	. . . . . . . . . . . . 13 |- ((((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) /\ (k e. NN /\ j <_ k)) -> (sup(ran F, RR, < ) - x) < (F` j))
68		climsup.2	. . . . . . . . . . . . . . . . 17 |- (k e. NN -> (F` k) <_ (F` (k + 1)))
69	1, 68	monoord 7523	. . . . . . . . . . . . . . . 16 |- ((j e. NN /\ k e. NN /\ j <_ k) -> (F` j) <_ (F` k))
70	69	3expb 1068	. . . . . . . . . . . . . . 15 |- ((j e. NN /\ (k e. NN /\ j <_ k)) -> (F` j) <_ (F` k))
71	70	adantlr 429	. . . . . . . . . . . . . 14 |- (((j e. NN /\ (F` j) = y) /\ (k e. NN /\ j <_ k)) -> (F` j) <_ (F` k))
72	71	adantll 428	. . . . . . . . . . . . 13 |- ((((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) /\ (k e. NN /\ j <_ k)) -> (F` j) <_ (F` k))
73	58, 61, 63, 67, 72	ltletrd 6698	. . . . . . . . . . . 12 |- ((((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) /\ (k e. NN /\ j <_ k)) -> (sup(ran F, RR, < ) - x) < (F` k))
74		recn 6466	. . . . . . . . . . . . . . . . . . 19 |- ((F` k) e. RR -> (F` k) e. CC)
75		abssub 8146	. . . . . . . . . . . . . . . . . . . 20 |- (((F` k) e. CC /\ sup(ran F, RR, < ) e. CC) -> (abs` ((F` k) - sup(ran F, RR, < ))) = (abs` (sup(ran F, RR, < ) - (F` k))))
76	34, 75	mpan2 760	. . . . . . . . . . . . . . . . . . 19 |- ((F` k) e. CC -> (abs` ((F` k) - sup(ran F, RR, < ))) = (abs` (sup(ran F, RR, < ) - (F` k))))
77	62, 74, 76	3syl 24	. . . . . . . . . . . . . . . . . 18 |- (k e. NN -> (abs` ((F` k) - sup(ran F, RR, < ))) = (abs` (sup(ran F, RR, < ) - (F` k))))
78		resubcl 6601	. . . . . . . . . . . . . . . . . . . 20 |- ((sup(ran F, RR, < ) e. RR /\ (F` k) e. RR) -> (sup(ran F, RR, < ) - (F` k)) e. RR)
79	78, 33, 62	sylancr 526	. . . . . . . . . . . . . . . . . . 19 |- (k e. NN -> (sup(ran F, RR, < ) - (F` k)) e. RR)
80		fnfvelrn 4786	. . . . . . . . . . . . . . . . . . . . . 22 |- ((F Fn NN /\ k e. NN) -> (F` k) e. ran F)
81	25, 80	mpan 759	. . . . . . . . . . . . . . . . . . . . 21 |- (k e. NN -> (F` k) e. ran F)
82	32	suprubii 7271	. . . . . . . . . . . . . . . . . . . . 21 |- ((F` k) e. ran F -> (F` k) <_ sup(ran F, RR, < ))
83	81, 82	syl 12	. . . . . . . . . . . . . . . . . . . 20 |- (k e. NN -> (F` k) <_ sup(ran F, RR, < ))
84		subge0 6863	. . . . . . . . . . . . . . . . . . . . 21 |- ((sup(ran F, RR, < ) e. RR /\ (F` k) e. RR) -> (0 <_ (sup(ran F, RR, < ) - (F` k)) <-> (F` k) <_ sup(ran F, RR, < )))
85	84, 33, 62	sylancr 526	. . . . . . . . . . . . . . . . . . . 20 |- (k e. NN -> (0 <_ (sup(ran F, RR, < ) - (F` k)) <-> (F` k) <_ sup(ran F, RR, < )))
86	83, 85	mpbird 213	. . . . . . . . . . . . . . . . . . 19 |- (k e. NN -> 0 <_ (sup(ran F, RR, < ) - (F` k)))
87		absid 8113	. . . . . . . . . . . . . . . . . . 19 |- (((sup(ran F, RR, < ) - (F` k)) e. RR /\ 0 <_ (sup(ran F, RR, < ) - (F` k))) -> (abs` (sup(ran F, RR, < ) - (F` k))) = (sup(ran F, RR, < ) - (F` k)))
88	79, 86, 87	syl11anc 524	. . . . . . . . . . . . . . . . . 18 |- (k e. NN -> (abs` (sup(ran F, RR, < ) - (F` k))) = (sup(ran F, RR, < ) - (F` k)))
89	77, 88	eqtrd 1925	. . . . . . . . . . . . . . . . 17 |- (k e. NN -> (abs` ((F` k) - sup(ran F, RR, < ))) = (sup(ran F, RR, < ) - (F` k)))
90	89	breq1d 3348	. . . . . . . . . . . . . . . 16 |- (k e. NN -> ((abs` ((F` k) - sup(ran F, RR, < ))) < x <-> (sup(ran F, RR, < ) - (F` k)) < x))
91	90	adantl 424	. . . . . . . . . . . . . . 15 |- ((x e. RR /\ k e. NN) -> ((abs` ((F` k) - sup(ran F, RR, < ))) < x <-> (sup(ran F, RR, < ) - (F` k)) < x))
92		ltsub23 6825	. . . . . . . . . . . . . . . . 17 |- ((sup(ran F, RR, < ) e. RR /\ x e. RR /\ (F` k) e. RR) -> ((sup(ran F, RR, < ) - x) < (F` k) <-> (sup(ran F, RR, < ) - (F` k)) < x))
93	33, 92	mp3an1 1178	. . . . . . . . . . . . . . . 16 |- ((x e. RR /\ (F` k) e. RR) -> ((sup(ran F, RR, < ) - x) < (F` k) <-> (sup(ran F, RR, < ) - (F` k)) < x))
94	93, 62	sylan2 500	. . . . . . . . . . . . . . 15 |- ((x e. RR /\ k e. NN) -> ((sup(ran F, RR, < ) - x) < (F` k) <-> (sup(ran F, RR, < ) - (F` k)) < x))
95	91, 94	bitr4d 590	. . . . . . . . . . . . . 14 |- ((x e. RR /\ k e. NN) -> ((abs` ((F` k) - sup(ran F, RR, < ))) < x <-> (sup(ran F, RR, < ) - x) < (F` k)))
96	95	ad2ant2r 445	. . . . . . . . . . . . 13 |- (((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (k e. NN /\ j <_ k)) -> ((abs` ((F` k) - sup(ran F, RR, < ))) < x <-> (sup(ran F, RR, < ) - x) < (F` k)))
97	96	adantlr 429	. . . . . . . . . . . 12 |- ((((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) /\ (k e. NN /\ j <_ k)) -> ((abs` ((F` k) - sup(ran F, RR, < ))) < x <-> (sup(ran F, RR, < ) - x) < (F` k)))
98	73, 97	mpbird 213	. . . . . . . . . . 11 |- ((((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) /\ (k e. NN /\ j <_ k)) -> (abs` ((F` k) - sup(ran F, RR, < ))) < x)
99	98	exp32 408	. . . . . . . . . 10 |- (((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) -> (k e. NN -> (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x)))
100	99	r19.21aiv 2175	. . . . . . . . 9 |- (((x e. RR /\ (sup(ran F, RR, < ) - x) < y) /\ (j e. NN /\ (F` j) = y)) -> A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x))
101	100	exp32 408	. . . . . . . 8 |- ((x e. RR /\ (sup(ran F, RR, < ) - x) < y) -> (j e. NN -> ((F` j) = y -> A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x))))
102	101	reximdvai 2201	. . . . . . 7 |- ((x e. RR /\ (sup(ran F, RR, < ) - x) < y) -> (E.j e. NN (F` j) = y -> E.j e. NN A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x)))
103		fvelrnb 4719	. . . . . . . 8 |- (F Fn NN -> (y e. ran F <-> E.j e. NN (F` j) = y))
104	25, 103	ax-mp 7	. . . . . . 7 |- (y e. ran F <-> E.j e. NN (F` j) = y)
105	102, 104	syl5ib 223	. . . . . 6 |- ((x e. RR /\ (sup(ran F, RR, < ) - x) < y) -> (y e. ran F -> E.j e. NN A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x)))
106	105	ex 402	. . . . 5 |- (x e. RR -> ((sup(ran F, RR, < ) - x) < y -> (y e. ran F -> E.j e. NN A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x))))
107	106	com23 36	. . . 4 |- (x e. RR -> (y e. ran F -> ((sup(ran F, RR, < ) - x) < y -> E.j e. NN A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x))))
108	107	r19.23adv 2215	. . 3 |- (x e. RR -> (E.y e. ran F(sup(ran F, RR, < ) - x) < y -> E.j e. NN A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x)))
109	56, 108	syld 30	. 2 |- (x e. RR -> (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` ((F` k) - sup(ran F, RR, < ))) < x)))
110	36, 109	mprgbir 2163	1 |- F ~~> sup(ran F, RR, < )

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875   class class class wbr 3338  dom cdm 3986  ran crn 3987   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  supcsup 5663  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   - cmin 6445   <_ cle 6448  NNcn 6449   < clt 6653  abscabs 8000   ~~> cli 8234

This theorem is referenced by:  cvgcmp2lem 8440

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-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-z 7345  df-uz 7587  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235

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