Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > limcres | Structured version Visualization version Unicode version |
Description: If is an interior point of relative to the domain , then a limit point of extends to a limit of . (Contributed by Mario Carneiro, 27-Dec-2016.) |
Ref | Expression |
---|---|
limcres.f | |
limcres.c | |
limcres.a | |
limcres.k | ℂfld |
limcres.j | ↾t |
limcres.i |
Ref | Expression |
---|---|
limcres | lim lim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcrcl 23638 | . . . . . 6 lim | |
2 | 1 | simp3d 1075 | . . . . 5 lim |
3 | limccl 23639 | . . . . . 6 lim | |
4 | 3 | sseli 3599 | . . . . 5 lim |
5 | 2, 4 | jca 554 | . . . 4 lim |
6 | 5 | a1i 11 | . . 3 lim |
7 | limcrcl 23638 | . . . . . 6 lim | |
8 | 7 | simp3d 1075 | . . . . 5 lim |
9 | limccl 23639 | . . . . . 6 lim | |
10 | 9 | sseli 3599 | . . . . 5 lim |
11 | 8, 10 | jca 554 | . . . 4 lim |
12 | 11 | a1i 11 | . . 3 lim |
13 | limcres.j | . . . . . . . 8 ↾t | |
14 | limcres.k | . . . . . . . . . 10 ℂfld | |
15 | 14 | cnfldtopon 22586 | . . . . . . . . 9 TopOn |
16 | limcres.a | . . . . . . . . . . 11 | |
17 | 16 | adantr 481 | . . . . . . . . . 10 |
18 | simprl 794 | . . . . . . . . . . 11 | |
19 | 18 | snssd 4340 | . . . . . . . . . 10 |
20 | 17, 19 | unssd 3789 | . . . . . . . . 9 |
21 | resttopon 20965 | . . . . . . . . 9 TopOn ↾t TopOn | |
22 | 15, 20, 21 | sylancr 695 | . . . . . . . 8 ↾t TopOn |
23 | 13, 22 | syl5eqel 2705 | . . . . . . 7 TopOn |
24 | topontop 20718 | . . . . . . 7 TopOn | |
25 | 23, 24 | syl 17 | . . . . . 6 |
26 | limcres.c | . . . . . . . . 9 | |
27 | 26 | adantr 481 | . . . . . . . 8 |
28 | unss1 3782 | . . . . . . . 8 | |
29 | 27, 28 | syl 17 | . . . . . . 7 |
30 | toponuni 20719 | . . . . . . . 8 TopOn | |
31 | 23, 30 | syl 17 | . . . . . . 7 |
32 | 29, 31 | sseqtrd 3641 | . . . . . 6 |
33 | limcres.i | . . . . . . 7 | |
34 | 33 | adantr 481 | . . . . . 6 |
35 | elun 3753 | . . . . . . . . 9 | |
36 | simplrr 801 | . . . . . . . . . . 11 | |
37 | limcres.f | . . . . . . . . . . . . 13 | |
38 | 37 | adantr 481 | . . . . . . . . . . . 12 |
39 | 38 | ffvelrnda 6359 | . . . . . . . . . . 11 |
40 | 36, 39 | ifcld 4131 | . . . . . . . . . 10 |
41 | elsni 4194 | . . . . . . . . . . . . 13 | |
42 | 41 | adantl 482 | . . . . . . . . . . . 12 |
43 | 42 | iftrued 4094 | . . . . . . . . . . 11 |
44 | simplrr 801 | . . . . . . . . . . 11 | |
45 | 43, 44 | eqeltrd 2701 | . . . . . . . . . 10 |
46 | 40, 45 | jaodan 826 | . . . . . . . . 9 |
47 | 35, 46 | sylan2b 492 | . . . . . . . 8 |
48 | eqid 2622 | . . . . . . . 8 | |
49 | 47, 48 | fmptd 6385 | . . . . . . 7 |
50 | 31 | feq2d 6031 | . . . . . . 7 |
51 | 49, 50 | mpbid 222 | . . . . . 6 |
52 | eqid 2622 | . . . . . . 7 | |
53 | 15 | toponunii 20721 | . . . . . . 7 |
54 | 52, 53 | cnprest 21093 | . . . . . 6 ↾t |
55 | 25, 32, 34, 51, 54 | syl22anc 1327 | . . . . 5 ↾t |
56 | 13, 14, 48, 38, 17, 18 | ellimc 23637 | . . . . 5 lim |
57 | eqid 2622 | . . . . . . 7 ↾t ↾t | |
58 | eqid 2622 | . . . . . . 7 | |
59 | 38, 27 | fssresd 6071 | . . . . . . 7 |
60 | 27, 17 | sstrd 3613 | . . . . . . 7 |
61 | 57, 14, 58, 59, 60, 18 | ellimc 23637 | . . . . . 6 lim ↾t |
62 | 29 | resmptd 5452 | . . . . . . . 8 |
63 | elun 3753 | . . . . . . . . . . 11 | |
64 | velsn 4193 | . . . . . . . . . . . 12 | |
65 | 64 | orbi2i 541 | . . . . . . . . . . 11 |
66 | 63, 65 | bitri 264 | . . . . . . . . . 10 |
67 | pm5.61 749 | . . . . . . . . . . . 12 | |
68 | fvres 6207 | . . . . . . . . . . . . 13 | |
69 | 68 | adantr 481 | . . . . . . . . . . . 12 |
70 | 67, 69 | sylbi 207 | . . . . . . . . . . 11 |
71 | 70 | ifeq2da 4117 | . . . . . . . . . 10 |
72 | 66, 71 | sylbi 207 | . . . . . . . . 9 |
73 | 72 | mpteq2ia 4740 | . . . . . . . 8 |
74 | 62, 73 | syl6reqr 2675 | . . . . . . 7 |
75 | 13 | oveq1i 6660 | . . . . . . . . . 10 ↾t ↾t ↾t |
76 | 15 | a1i 11 | . . . . . . . . . . 11 TopOn |
77 | cnex 10017 | . . . . . . . . . . . . 13 | |
78 | 77 | ssex 4802 | . . . . . . . . . . . 12 |
79 | 20, 78 | syl 17 | . . . . . . . . . . 11 |
80 | restabs 20969 | . . . . . . . . . . 11 TopOn ↾t ↾t ↾t | |
81 | 76, 29, 79, 80 | syl3anc 1326 | . . . . . . . . . 10 ↾t ↾t ↾t |
82 | 75, 81 | syl5req 2669 | . . . . . . . . 9 ↾t ↾t |
83 | 82 | oveq1d 6665 | . . . . . . . 8 ↾t ↾t |
84 | 83 | fveq1d 6193 | . . . . . . 7 ↾t ↾t |
85 | 74, 84 | eleq12d 2695 | . . . . . 6 ↾t ↾t |
86 | 61, 85 | bitrd 268 | . . . . 5 lim ↾t |
87 | 55, 56, 86 | 3bitr4rd 301 | . . . 4 lim lim |
88 | 87 | ex 450 | . . 3 lim lim |
89 | 6, 12, 88 | pm5.21ndd 369 | . 2 lim lim |
90 | 89 | eqrdv 2620 | 1 lim lim |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 cvv 3200 cun 3572 wss 3574 cif 4086 csn 4177 cuni 4436 cmpt 4729 cdm 5114 cres 5116 wf 5884 cfv 5888 (class class class)co 6650 cc 9934 ↾t crest 16081 ctopn 16082 ℂfldccnfld 19746 ctop 20698 TopOnctopon 20715 cnt 20821 ccnp 21029 lim climc 23626 |
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-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-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 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-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-ntr 20824 df-cnp 21032 df-xms 22125 df-ms 22126 df-limc 23630 |
This theorem is referenced by: dvreslem 23673 dvaddbr 23701 dvmulbr 23702 lhop2 23778 lhop 23779 limciccioolb 39853 limcicciooub 39869 limcresiooub 39874 limcresioolb 39875 ioccncflimc 40098 icocncflimc 40102 dirkercncflem3 40322 fourierdlem32 40356 fourierdlem33 40357 fourierdlem48 40371 fourierdlem49 40372 fourierdlem62 40385 fouriersw 40448 |
Copyright terms: Public domain | W3C validator |