climxrre - Mathbox for Glauco Siliprandi

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Theorem climxrre 39982

Description: If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis

is probably not enough, since in principle we could have

and

). (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
climxrre.m
climxrre.z
climxrre.f
climxrre.a
climxrre.c

**Assertion**
Ref	Expression
climxrre

Distinct variable groups:

Proof of Theorem climxrre Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climxrre.m	. . . . 5
2	1	ad2antrr 762	. . . 4 $/\$ $/\$
3		climxrre.z	. . . 4
4		climxrre.f	. . . . 5
5	4	ad2antrr 762	. . . 4 $/\$ $/\$
6		climxrre.c	. . . . 5
7	6	ad2antrr 762	. . . 4 $/\$ $/\$
8		simpr 477	. . . . . . . 8 $/\$
9		climxrre.a	. . . . . . . . . 10
10	9	recnd 10068	. . . . . . . . 9
11	10	adantr 481	. . . . . . . 8 $/\$
12	8, 11	subcld 10392	. . . . . . 7 $/\$
13		renepnf 10087	. . . . . . . . . . 11
14	13	necomd 2849	. . . . . . . . . 10
15	9, 14	syl 17	. . . . . . . . 9
16	15	adantr 481	. . . . . . . 8 $/\$
17	8, 11, 16	subne0d 10401	. . . . . . 7 $/\$
18	12, 17	absrpcld 14187	. . . . . 6 $/\$
19	18	adantr 481	. . . . 5 $/\$ $/\$
20		simpr 477	. . . . . . . 8 $/\$
21	10	adantr 481	. . . . . . . 8 $/\$
22	20, 21	subcld 10392	. . . . . . 7 $/\$
23	9	adantr 481	. . . . . . . . 9 $/\$
24		renemnf 10088	. . . . . . . . . 10
25	24	necomd 2849	. . . . . . . . 9
26	23, 25	syl 17	. . . . . . . 8 $/\$
27	20, 21, 26	subne0d 10401	. . . . . . 7 $/\$
28	22, 27	absrpcld 14187	. . . . . 6 $/\$
29	28	adantlr 751	. . . . 5 $/\$ $/\$
30	19, 29	ifcld 4131	. . . 4 $/\$ $/\$
31	19	rpred 11872	. . . . . 6 $/\$ $/\$
32	29	rpred 11872	. . . . . 6 $/\$ $/\$
33	31, 32	min1d 39702	. . . . 5 $/\$ $/\$
34	33	adantr 481	. . . 4 $/\$ $/\$ $/\$
35	31, 32	min2d 39703	. . . . 5 $/\$ $/\$
36	35	adantr 481	. . . 4 $/\$ $/\$ $/\$
37	2, 3, 5, 7, 30, 34, 36	climxrrelem 39981	. . 3 $/\$ $/\$
38	1	ad2antrr 762	. . . 4 $/\$ $/\$
39	4	ad2antrr 762	. . . 4 $/\$ $/\$
40	6	ad2antrr 762	. . . 4 $/\$ $/\$
41	18	adantr 481	. . . 4 $/\$ $/\$
42	18	rpred 11872	. . . . . 6 $/\$
43	42	leidd 10594	. . . . 5 $/\$
44	43	ad2antrr 762	. . . 4 $/\$ $/\$ $/\$
45		pm2.21 120	. . . . . 6
46	45	imp 445	. . . . 5 $/\$
47	46	adantll 750	. . . 4 $/\$ $/\$ $/\$
48	38, 3, 39, 40, 41, 44, 47	climxrrelem 39981	. . 3 $/\$ $/\$
49	37, 48	pm2.61dan 832	. 2 $/\$
50	1	ad2antrr 762	. . . 4 $/\$ $/\$
51	4	ad2antrr 762	. . . 4 $/\$ $/\$
52	6	ad2antrr 762	. . . 4 $/\$ $/\$
53	28	adantlr 751	. . . 4 $/\$ $/\$
54		pm2.21 120	. . . . . 6
55	54	imp 445	. . . . 5 $/\$
56	55	ad4ant24 1298	. . . 4 $/\$ $/\$ $/\$
57	28	rpred 11872	. . . . . 6 $/\$
58	57	leidd 10594	. . . . 5 $/\$
59	58	ad4ant13 1292	. . . 4 $/\$ $/\$ $/\$
60	50, 3, 51, 52, 53, 56, 59	climxrrelem 39981	. . 3 $/\$ $/\$
61		nfv 1843	. . . . . . 7 $/\$ $/\$
62		nfv 1843	. . . . . . . 8
63		nfra1 2941	. . . . . . . 8
64	62, 63	nfan 1828	. . . . . . 7 $/\$
65	61, 64	nfan 1828	. . . . . 6 $/\$ $/\$ $/\$ $/\$
66		simp-4l 806	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
67	3	uztrn2 11705	. . . . . . . . . 10 $/\$
68	67	adantlr 751	. . . . . . . . 9 $/\$ $/\$
69	68	adantll 750	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
70		simpr 477	. . . . . . . . 9 $/\$
71	4	fdmd 39420	. . . . . . . . . 10
72	71	adantr 481	. . . . . . . . 9 $/\$
73	70, 72	eleqtrrd 2704	. . . . . . . 8 $/\$
74	66, 69, 73	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
75	4	ffvelrnda 6359	. . . . . . . . 9 $/\$
76	66, 69, 75	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
77		rspa 2930	. . . . . . . . . . 11 $/\$
78	77	adantll 750	. . . . . . . . . 10 $/\$ $/\$
79	78	adantll 750	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
80		simpllr 799	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
81		nelne2 2891	. . . . . . . . 9 $/\$
82	79, 80, 81	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
83		simp-4r 807	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
84		nelne2 2891	. . . . . . . . 9 $/\$
85	79, 83, 84	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
86	76, 82, 85	xrred 39581	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
87	74, 86	jca 554	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
88	65, 87	ralrimia 39315	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
89	4	ffund 6049	. . . . . . 7
90		ffvresb 6394	. . . . . . 7 $/\$
91	89, 90	syl 17	. . . . . 6 $/\$
92	91	ad3antrrr 766	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
93	88, 92	mpbird 247	. . . 4 $/\$ $/\$ $/\$ $/\$
94		r19.26 3064	. . . . . . . . 9 $/\$ $/\$
95	94	simplbi 476	. . . . . . . 8 $/\$
96	95	ad2antll 765	. . . . . . 7 $/\$ $/\$ $/\$
97		breq2 4657	. . . . . . . . . 10
98	97	anbi2d 740	. . . . . . . . 9 $/\$ $/\$
99	98	rexralbidv 3058	. . . . . . . 8 $/\$ $/\$
100	3	fvexi 6202	. . . . . . . . . . . . 13
101	100	a1i 11	. . . . . . . . . . . 12
102	4, 101	fexd 39296	. . . . . . . . . . 11
103		eqidd 2623	. . . . . . . . . . 11 $/\$
104	102, 103	clim 14225	. . . . . . . . . 10 $/\$ $/\$
105	6, 104	mpbid 222	. . . . . . . . 9 $/\$ $/\$
106	105	simprd 479	. . . . . . . 8 $/\$
107		1rp 11836	. . . . . . . . 9
108	107	a1i 11	. . . . . . . 8
109	99, 106, 108	rspcdva 3316	. . . . . . 7 $/\$
110	96, 109	reximddv 3018	. . . . . 6
111	3	rexuz3 14088	. . . . . . 7
112	1, 111	syl 17	. . . . . 6
113	110, 112	mpbird 247	. . . . 5
114	113	ad2antrr 762	. . . 4 $/\$ $/\$
115	93, 114	reximddv 3018	. . 3 $/\$ $/\$
116	60, 115	pm2.61dan 832	. 2 $/\$
117	49, 116	pm2.61dan 832	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913

cvv 3200

cif 4086 class class class wbr 4653

cdm 5114

cres 5116

wfun 5882

wf 5884

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

c1 9937

cpnf 10071

cmnf 10072

cxr 10073

clt 10074

cle 10075

cmin 10266

cz 11377

cuz 11687

crp 11832

cabs 13974

cli 14215

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-rep 4771 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-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-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219

This theorem is referenced by: xlimclim2 40066

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