limsupgre - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > limsupgre		Structured version Visualization version Unicode version

Theorem limsupgre 14212

Description: If a sequence of real numbers has upper bounded limit supremum, then all the partial suprema are real. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

**Hypotheses**
Ref	Expression
limsupval.1
limsupgre.z

**Assertion**
Ref	Expression
limsupgre	$/\$ $/\$

Distinct variable groups:

Allowed substitution hint:

(

)

Proof of Theorem limsupgre Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 11974	. . . 4
2	1	supex 8369	. . 3
3	2	a1i 11	. 2 $/\$ $/\$ $/\$
4		limsupval.1	. . 3
5	4	a1i 11	. 2 $/\$ $/\$
6	4	limsupgval 14207	. . . 4
7	6	adantl 482	. . 3 $/\$ $/\$ $/\$
8		simpl3 1066	. . . . . . 7 $/\$ $/\$ $/\$
9		limsupgre.z	. . . . . . . . . . 11
10		uzssz 11707	. . . . . . . . . . 11
11	9, 10	eqsstri 3635	. . . . . . . . . 10
12		zssre 11384	. . . . . . . . . 10
13	11, 12	sstri 3612	. . . . . . . . 9
14	13	a1i 11	. . . . . . . 8 $/\$ $/\$ $/\$
15		simpl2 1065	. . . . . . . . 9 $/\$ $/\$ $/\$
16		ressxr 10083	. . . . . . . . 9
17		fss 6056	. . . . . . . . 9 $/\$
18	15, 16, 17	sylancl 694	. . . . . . . 8 $/\$ $/\$ $/\$
19		pnfxr 10092	. . . . . . . . 9
20	19	a1i 11	. . . . . . . 8 $/\$ $/\$ $/\$
21	4	limsuplt 14210	. . . . . . . 8 $/\$ $/\$
22	14, 18, 20, 21	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$
23	8, 22	mpbid 222	. . . . . 6 $/\$ $/\$ $/\$
24		fzfi 12771	. . . . . . . 8
25	15	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
26		elfzuz 12338	. . . . . . . . . . 11
27	26, 9	syl6eleqr 2712	. . . . . . . . . 10
28		ffvelrn 6357	. . . . . . . . . 10 $/\$
29	25, 27, 28	syl2an 494	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29	ralrimiva 2966	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
31		fimaxre3 10970	. . . . . . . 8 $/\$
32	24, 30, 31	sylancr 695	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
33		simpr 477	. . . . . . . . . 10 $/\$ $/\$ $/\$
34	33	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	4	limsupgf 14206	. . . . . . . . . 10
36	35	ffvelrni 6358	. . . . . . . . 9
37	34, 36	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		simprl 794	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	16, 38	sseldi 3601	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		simprl 794	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
41	40	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	35	ffvelrni 6358	. . . . . . . . . 10
43	41, 42	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	39, 43	ifcld 4131	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	19	a1i 11	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	40	ad2antrr 762	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	13	a1i 11	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	47	sselda 3603	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		xrleid 11983	. . . . . . . . . . . . . . . . 17
50	43, 49	syl 17	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	18	ad2antrr 762	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	4	limsupgle 14208	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
53	47, 51, 41, 43, 52	syl211anc 1332	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	50, 53	mpbid 222	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	54	r19.21bi 2932	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	55	imp 445	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	46, 42	syl 17	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	39	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59		xrmax1 12006	. . . . . . . . . . . . . . . 16 $/\$
60	57, 58, 59	syl2anc 693	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	51	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	44	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		xrletr 11989	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
64	61, 57, 62, 63	syl3anc 1326	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	60, 64	mpan2d 710	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	65	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	56, 66	mpd 15	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68		simpr 477	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69	68, 9	syl6eleq 2711	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	41	flcld 12599	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	70	adantr 481	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72		elfz5 12334	. . . . . . . . . . . . . . . . 17 $/\$
73	69, 71, 72	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74	11, 68	sseldi 3601	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75		flge 12606	. . . . . . . . . . . . . . . . 17 $/\$
76	46, 74, 75	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77	73, 76	bitr4d 271	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	77	biimpar 502	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79		simprr 796	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
80	79	ad2antrr 762	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
81		fveq2 6191	. . . . . . . . . . . . . . . 16
82	81	breq1d 4663	. . . . . . . . . . . . . . 15
83	82	rspcv 3305	. . . . . . . . . . . . . 14
84	78, 80, 83	sylc 65	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
85		xrmax2 12007	. . . . . . . . . . . . . . . . 17 $/\$
86	43, 39, 85	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
87	86	adantr 481	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
88		xrletr 11989	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
89	61, 58, 62, 88	syl3anc 1326	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
90	87, 89	mpan2d 710	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
91	90	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
92	84, 91	mpd 15	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
93	46, 48, 67, 92	lecasei 10143	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
94	93	a1d 25	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
95	94	ralrimiva 2966	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
96	4	limsupgle 14208	. . . . . . . . . 10 $/\$ $/\$ $/\$
97	47, 51, 34, 44, 96	syl211anc 1332	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
98	95, 97	mpbird 247	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
99	38	ltpnfd 11955	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
100		simplrr 801	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
101		breq1 4656	. . . . . . . . . 10
102		breq1 4656	. . . . . . . . . 10
103	101, 102	ifboth 4124	. . . . . . . . 9 $/\$
104	99, 100, 103	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
105	37, 44, 45, 98, 104	xrlelttrd 11991	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
106	32, 105	rexlimddv 3035	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
107	23, 106	rexlimddv 3035	. . . . 5 $/\$ $/\$ $/\$
108	7, 107	eqbrtrrd 4677	. . . 4 $/\$ $/\$ $/\$
109		imassrn 5477	. . . . . . . . 9
110		frn 6053	. . . . . . . . . 10
111	15, 110	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$
112	109, 111	syl5ss 3614	. . . . . . . 8 $/\$ $/\$ $/\$
113	112, 16	syl6ss 3615	. . . . . . 7 $/\$ $/\$ $/\$
114		df-ss 3588	. . . . . . 7
115	113, 114	sylib 208	. . . . . 6 $/\$ $/\$ $/\$
116	115, 112	eqsstrd 3639	. . . . 5 $/\$ $/\$ $/\$
117		simpl1 1064	. . . . . . . . . . 11 $/\$ $/\$ $/\$
118		flcl 12596	. . . . . . . . . . . . . 14
119	118	adantl 482	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
120	119	peano2zd 11485	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
121	120, 117	ifcld 4131	. . . . . . . . . . 11 $/\$ $/\$ $/\$
122	117	zred 11482	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
123	120	zred 11482	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
124		max1 12016	. . . . . . . . . . . 12 $/\$
125	122, 123, 124	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$
126		eluz2 11693	. . . . . . . . . . 11 $/\$ $/\$
127	117, 121, 125, 126	syl3anbrc 1246	. . . . . . . . . 10 $/\$ $/\$ $/\$
128	127, 9	syl6eleqr 2712	. . . . . . . . 9 $/\$ $/\$ $/\$
129		fdm 6051	. . . . . . . . . 10
130	15, 129	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$
131	128, 130	eleqtrrd 2704	. . . . . . . 8 $/\$ $/\$ $/\$
132	121	zred 11482	. . . . . . . . 9 $/\$ $/\$ $/\$
133		fllep1 12602	. . . . . . . . . . 11
134	133	adantl 482	. . . . . . . . . 10 $/\$ $/\$ $/\$
135		max2 12018	. . . . . . . . . . 11 $/\$
136	122, 123, 135	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$
137	33, 123, 132, 134, 136	letrd 10194	. . . . . . . . 9 $/\$ $/\$ $/\$
138		elicopnf 12269	. . . . . . . . . 10 $/\$
139	138	adantl 482	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
140	132, 137, 139	mpbir2and 957	. . . . . . . 8 $/\$ $/\$ $/\$
141		inelcm 4032	. . . . . . . 8 $/\$
142	131, 140, 141	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$
143		imadisj 5484	. . . . . . . 8
144	143	necon3bii 2846	. . . . . . 7
145	142, 144	sylibr 224	. . . . . 6 $/\$ $/\$ $/\$
146	115, 145	eqnetrd 2861	. . . . 5 $/\$ $/\$ $/\$
147		supxrre1 12160	. . . . 5 $/\$
148	116, 146, 147	syl2anc 693	. . . 4 $/\$ $/\$ $/\$
149	108, 148	mpbird 247	. . 3 $/\$ $/\$ $/\$
150	7, 149	eqeltrd 2701	. 2 $/\$ $/\$ $/\$
151	3, 5, 150	fmpt2d 6393	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913

cvv 3200

cin 3573

wss 3574

c0 3915

cif 4086 class class class wbr 4653

cmpt 4729

cdm 5114

crn 5115

cima 5117

wf 5884

cfv 5888 (class class class)co 6650

cfn 7955

csup 8346

cr 9935

c1 9937

caddc 9939

cpnf 10071

cxr 10073

clt 10074

cle 10075

cz 11377

cuz 11687

cico 12177

cfz 12326

cfl 12591

clsp 14201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-ico 12181 df-fz 12327 df-fl 12593 df-limsup 14202

This theorem is referenced by: mbflimsup 23433

W3C validator