rlimuni - Metamath Proof Explorer

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Theorem rlimuni 14281

Description: A real function whose domain is unbounded above converges to at most one limit. (Contributed by Mario Carneiro, 8-May-2016.)

**Hypotheses**
Ref	Expression
rlimuni.1
rlimuni.2
rlimuni.3
rlimuni.4

**Assertion**
Ref	Expression
rlimuni

Proof of Theorem rlimuni Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimuni.3	. . . . . . . . . . . 12
2		rlimcl 14234	. . . . . . . . . . . 12
3	1, 2	syl 17	. . . . . . . . . . 11
4	3	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$
5		rlimuni.4	. . . . . . . . . . . 12
6		rlimcl 14234	. . . . . . . . . . . 12
7	5, 6	syl 17	. . . . . . . . . . 11
8	7	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$
9	4, 8	subcld 10392	. . . . . . . . 9 $/\$ $/\$
10	9	abscld 14175	. . . . . . . 8 $/\$ $/\$
11	10	ltnrd 10171	. . . . . . 7 $/\$ $/\$
12		rlimuni.1	. . . . . . . . . . . . . . . 16
13	12	ffvelrnda 6359	. . . . . . . . . . . . . . 15 $/\$
14	13	adantlr 751	. . . . . . . . . . . . . 14 $/\$ $/\$
15	14, 4	abssubd 14192	. . . . . . . . . . . . 13 $/\$ $/\$
16	15	breq1d 4663	. . . . . . . . . . . 12 $/\$ $/\$
17	16	anbi1d 741	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
18		abs3lem 14078	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
19	4, 8, 14, 10, 18	syl22anc 1327	. . . . . . . . . . 11 $/\$ $/\$ $/\$
20	17, 19	sylbid 230	. . . . . . . . . 10 $/\$ $/\$ $/\$
21	20	imim2d 57	. . . . . . . . 9 $/\$ $/\$ $/\$
22	21	com23 86	. . . . . . . 8 $/\$ $/\$ $/\$
23	22	impd 447	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
24	11, 23	mtod 189	. . . . . 6 $/\$ $/\$ $/\$ $/\$
25	24	nrexdv 3001	. . . . 5 $/\$ $/\$ $/\$
26		r19.29r 3073	. . . . 5 $/\$ $/\$ $/\$ $/\$
27	25, 26	nsyl 135	. . . 4 $/\$ $/\$ $/\$
28	27	nrexdv 3001	. . 3 $/\$ $/\$
29		rlimuni.2	. . . . 5
30		fdm 6051	. . . . . . . . 9
31	12, 30	syl 17	. . . . . . . 8
32		rlimss 14233	. . . . . . . . 9
33	1, 32	syl 17	. . . . . . . 8
34	31, 33	eqsstr3d 3640	. . . . . . 7
35		ressxr 10083	. . . . . . 7
36	34, 35	syl6ss 3615	. . . . . 6
37		supxrunb1 12149	. . . . . 6
38	36, 37	syl 17	. . . . 5
39	29, 38	mpbird 247	. . . 4
40		r19.29 3072	. . . . 5 $/\$ $/\$ $/\$ $/\$
41	40	ex 450	. . . 4 $/\$ $/\$ $/\$
42	39, 41	syl 17	. . 3 $/\$ $/\$ $/\$
43	28, 42	mtod 189	. 2 $/\$
44	12	adantr 481	. . . . . . 7 $/\$
45		ffvelrn 6357	. . . . . . . 8 $/\$
46	45	ralrimiva 2966	. . . . . . 7
47	44, 46	syl 17	. . . . . 6 $/\$
48	3	adantr 481	. . . . . . . . 9 $/\$
49	7	adantr 481	. . . . . . . . 9 $/\$
50	48, 49	subcld 10392	. . . . . . . 8 $/\$
51		simpr 477	. . . . . . . . 9 $/\$
52	48, 49, 51	subne0d 10401	. . . . . . . 8 $/\$
53	50, 52	absrpcld 14187	. . . . . . 7 $/\$
54	53	rphalfcld 11884	. . . . . 6 $/\$
55	44	feqmptd 6249	. . . . . . 7 $/\$
56	1	adantr 481	. . . . . . 7 $/\$
57	55, 56	eqbrtrrd 4677	. . . . . 6 $/\$
58	47, 54, 57	rlimi 14244	. . . . 5 $/\$
59	5	adantr 481	. . . . . . 7 $/\$
60	55, 59	eqbrtrrd 4677	. . . . . 6 $/\$
61	47, 54, 60	rlimi 14244	. . . . 5 $/\$
62	34	adantr 481	. . . . . 6 $/\$
63		rexanre 14086	. . . . . 6 $/\$ $/\$
64	62, 63	syl 17	. . . . 5 $/\$ $/\$ $/\$
65	58, 61, 64	mpbir2and 957	. . . 4 $/\$ $/\$
66	65	ex 450	. . 3 $/\$
67	66	necon1bd 2812	. 2 $/\$
68	43, 67	mpd 15	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

wss 3574 class class class wbr 4653

cmpt 4729

cdm 5114

wf 5884

cfv 5888 (class class class)co 6650

csup 8346

cc 9934

cr 9935

cpnf 10071

cxr 10073

clt 10074

cle 10075

cmin 10266

cdiv 10684

c2 11070

cabs 13974

crli 14216

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-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-pm 7860 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-rlim 14220

This theorem is referenced by: rlimdm 14282 rlimdmafv 41257

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