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Mirrors > Home > MPE Home > Th. List > cnextucn | Structured version Visualization version Unicode version |
Description: Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology on , a subset dense in , this states a condition for from to a space Hausdorff and complete to be extensible by continuity. (Contributed by Thierry Arnoux, 4-Dec-2017.) |
Ref | Expression |
---|---|
cnextucn.x | |
cnextucn.y | |
cnextucn.j | |
cnextucn.k | |
cnextucn.u | UnifSt |
cnextucn.v | |
cnextucn.t | |
cnextucn.w | CUnifSp |
cnextucn.h | |
cnextucn.a | |
cnextucn.f | |
cnextucn.c | |
cnextucn.l | ↾t CauFilu |
Ref | Expression |
---|---|
cnextucn | CnExt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . 2 | |
2 | eqid 2622 | . 2 | |
3 | cnextucn.v | . . 3 | |
4 | cnextucn.j | . . . 4 | |
5 | 4 | tpstop 20741 | . . 3 |
6 | 3, 5 | syl 17 | . 2 |
7 | cnextucn.h | . 2 | |
8 | cnextucn.f | . . 3 | |
9 | cnextucn.t | . . . . 5 | |
10 | cnextucn.y | . . . . . 6 | |
11 | cnextucn.k | . . . . . 6 | |
12 | 10, 11 | tpsuni 20740 | . . . . 5 |
13 | 9, 12 | syl 17 | . . . 4 |
14 | 13 | feq3d 6032 | . . 3 |
15 | 8, 14 | mpbid 222 | . 2 |
16 | cnextucn.a | . . 3 | |
17 | cnextucn.x | . . . . 5 | |
18 | 17, 4 | tpsuni 20740 | . . . 4 |
19 | 3, 18 | syl 17 | . . 3 |
20 | 16, 19 | sseqtrd 3641 | . 2 |
21 | cnextucn.c | . . 3 | |
22 | 21, 19 | eqtrd 2656 | . 2 |
23 | 10, 11 | istps 20738 | . . . . . 6 TopOn |
24 | 9, 23 | sylib 208 | . . . . 5 TopOn |
25 | 24 | adantr 481 | . . . 4 TopOn |
26 | 19 | eleq2d 2687 | . . . . . . 7 |
27 | 26 | biimpar 502 | . . . . . 6 |
28 | 21 | adantr 481 | . . . . . 6 |
29 | 27, 28 | eleqtrrd 2704 | . . . . 5 |
30 | 1 | toptopon 20722 | . . . . . . . . 9 TopOn |
31 | 6, 30 | sylib 208 | . . . . . . . 8 TopOn |
32 | fveq2 6191 | . . . . . . . . . 10 TopOn TopOn | |
33 | 32 | eleq2d 2687 | . . . . . . . . 9 TopOn TopOn |
34 | 19, 33 | syl 17 | . . . . . . . 8 TopOn TopOn |
35 | 31, 34 | mpbird 247 | . . . . . . 7 TopOn |
36 | 35 | adantr 481 | . . . . . 6 TopOn |
37 | 16 | adantr 481 | . . . . . 6 |
38 | trnei 21696 | . . . . . 6 TopOn ↾t | |
39 | 36, 37, 27, 38 | syl3anc 1326 | . . . . 5 ↾t |
40 | 29, 39 | mpbid 222 | . . . 4 ↾t |
41 | 8 | adantr 481 | . . . 4 |
42 | flfval 21794 | . . . 4 TopOn ↾t ↾t ↾t | |
43 | 25, 40, 41, 42 | syl3anc 1326 | . . 3 ↾t ↾t |
44 | cnextucn.w | . . . . 5 CUnifSp | |
45 | 44 | adantr 481 | . . . 4 CUnifSp |
46 | cnextucn.l | . . . . . 6 ↾t CauFilu | |
47 | 27, 46 | syldan 487 | . . . . 5 ↾t CauFilu |
48 | cnextucn.u | . . . . . 6 UnifSt | |
49 | 48 | fveq2i 6194 | . . . . 5 CauFilu CauFiluUnifSt |
50 | 47, 49 | syl6eleq 2711 | . . . 4 ↾t CauFiluUnifSt |
51 | fvex 6201 | . . . . . . 7 | |
52 | 10, 51 | eqeltri 2697 | . . . . . 6 |
53 | 52 | a1i 11 | . . . . 5 |
54 | filfbas 21652 | . . . . . 6 ↾t ↾t | |
55 | 40, 54 | syl 17 | . . . . 5 ↾t |
56 | fmfil 21748 | . . . . 5 ↾t ↾t | |
57 | 53, 55, 41, 56 | syl3anc 1326 | . . . 4 ↾t |
58 | 10, 11 | cuspcvg 22105 | . . . 4 CUnifSp ↾t CauFiluUnifSt ↾t ↾t |
59 | 45, 50, 57, 58 | syl3anc 1326 | . . 3 ↾t |
60 | 43, 59 | eqnetrd 2861 | . 2 ↾t |
61 | cuspusp 22104 | . . . 4 CUnifSp UnifSp | |
62 | 44, 61 | syl 17 | . . 3 UnifSp |
63 | 11 | uspreg 22078 | . . 3 UnifSp |
64 | 62, 7, 63 | syl2anc 693 | . 2 |
65 | 1, 2, 6, 7, 15, 20, 22, 60, 64 | cnextcn 21871 | 1 CnExt |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 wss 3574 c0 3915 csn 4177 cuni 4436 wf 5884 cfv 5888 (class class class)co 6650 cbs 15857 ↾t crest 16081 ctopn 16082 cfbas 19734 ctop 20698 TopOnctopon 20715 ctps 20736 ccl 20822 cnei 20901 ccn 21028 cha 21112 creg 21113 cfil 21649 cfm 21737 cflim 21738 cflf 21739 CnExtccnext 21863 UnifStcuss 22057 UnifSpcusp 22058 CauFiluccfilu 22090 CUnifSpccusp 22101 |
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 |
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-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-iin 4523 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-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-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-cnp 21032 df-haus 21119 df-reg 21120 df-tx 21365 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-cnext 21864 df-ust 22004 df-utop 22035 df-usp 22061 df-cusp 22102 |
This theorem is referenced by: ucnextcn 22108 |
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