limsupbnd2 - Metamath Proof Explorer

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Theorem limsupbnd2 14214

Description: If a sequence is eventually greater than

, then the limsup is also greater than

. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

**Hypotheses**
Ref	Expression
limsupbnd.1
limsupbnd.2
limsupbnd.3
limsupbnd2.4
limsupbnd2.5

**Assertion**
Ref	Expression
limsupbnd2

Distinct variable groups:

Proof of Theorem limsupbnd2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupbnd2.5	. . 3
2		limsupbnd2.4	. . . . . . . . 9
3		limsupbnd.1	. . . . . . . . . . 11
4		ressxr 10083	. . . . . . . . . . 11
5	3, 4	syl6ss 3615	. . . . . . . . . 10
6		supxrunb1 12149	. . . . . . . . . 10
7	5, 6	syl 17	. . . . . . . . 9
8	2, 7	mpbird 247	. . . . . . . 8
9		ifcl 4130	. . . . . . . 8 $/\$
10		breq1 4656	. . . . . . . . . 10
11	10	rexbidv 3052	. . . . . . . . 9
12	11	rspccva 3308	. . . . . . . 8 $/\$
13	8, 9, 12	syl2an 494	. . . . . . 7 $/\$ $/\$
14		r19.29 3072	. . . . . . . 8 $/\$ $/\$
15		simplrr 801	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
16		simprl 794	. . . . . . . . . . . . . . . 16 $/\$ $/\$
17	16	adantr 481	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
18		max1 12016	. . . . . . . . . . . . . . 15 $/\$
19	15, 17, 18	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
20	17, 15, 9	syl2anc 693	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
21	3	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ $/\$
22	21	sselda 3603	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
23		letr 10131	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
24	15, 20, 22, 23	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
25	19, 24	mpand 711	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
26	25	imim1d 82	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
27	26	impd 447	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
28		max2 12018	. . . . . . . . . . . . . . 15 $/\$
29	15, 17, 28	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
30		letr 10131	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
31	17, 20, 22, 30	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	29, 31	mpand 711	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
33	32	adantld 483	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
34		eqid 2622	. . . . . . . . . . . . . . . . . . 19
35	34	limsupgf 14206	. . . . . . . . . . . . . . . . . 18
36	35	ffvelrni 6358	. . . . . . . . . . . . . . . . 17
37	36	adantl 482	. . . . . . . . . . . . . . . 16 $/\$
38		xrleid 11983	. . . . . . . . . . . . . . . 16
39	37, 38	syl 17	. . . . . . . . . . . . . . 15 $/\$
40	39	adantrr 753	. . . . . . . . . . . . . 14 $/\$ $/\$
41		limsupbnd.2	. . . . . . . . . . . . . . . 16
42	41	adantr 481	. . . . . . . . . . . . . . 15 $/\$ $/\$
43	16, 36	syl 17	. . . . . . . . . . . . . . 15 $/\$ $/\$
44	34	limsupgle 14208	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
45	21, 42, 16, 43, 44	syl211anc 1332	. . . . . . . . . . . . . 14 $/\$ $/\$
46	40, 45	mpbid 222	. . . . . . . . . . . . 13 $/\$ $/\$
47	46	r19.21bi 2932	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
48	33, 47	syld 47	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
49	27, 48	jcad 555	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
50		limsupbnd.3	. . . . . . . . . . . 12
51	50	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$
52	42	ffvelrnda 6359	. . . . . . . . . . 11 $/\$ $/\$ $/\$
53	43	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$
54		xrletr 11989	. . . . . . . . . . 11 $/\$ $/\$ $/\$
55	51, 52, 53, 54	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
56	49, 55	syld 47	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
57	56	rexlimdva 3031	. . . . . . . 8 $/\$ $/\$ $/\$
58	14, 57	syl5 34	. . . . . . 7 $/\$ $/\$ $/\$
59	13, 58	mpan2d 710	. . . . . 6 $/\$ $/\$
60	59	anassrs 680	. . . . 5 $/\$ $/\$
61	60	rexlimdva 3031	. . . 4 $/\$
62	61	ralrimdva 2969	. . 3
63	1, 62	mpd 15	. 2
64	34	limsuple 14209	. . 3 $/\$ $/\$
65	3, 41, 50, 64	syl3anc 1326	. 2
66	63, 65	mpbird 247	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

wrex 2913

cin 3573

wss 3574

cif 4086 class class class wbr 4653

cmpt 4729

cima 5117

wf 5884

cfv 5888 (class class class)co 6650

csup 8346

cr 9935

cpnf 10071

cxr 10073

clt 10074

cle 10075

cico 12177

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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 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-ico 12181 df-limsup 14202

This theorem is referenced by: caucvgrlem 14403 limsupre 39873

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