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Mirrors > Home > MPE Home > Th. List > ressusp | Structured version Visualization version Unicode version |
Description: The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Ref | Expression |
---|---|
ressusp.1 | |
ressusp.2 |
Ref | Expression |
---|---|
ressusp | UnifSp ↾s UnifSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressuss 22067 | . . . . 5 UnifSt ↾s UnifSt ↾t | |
2 | 1 | 3ad2ant3 1084 | . . . 4 UnifSp UnifSt ↾s UnifSt ↾t |
3 | simp1 1061 | . . . . . . 7 UnifSp UnifSp | |
4 | ressusp.1 | . . . . . . . 8 | |
5 | eqid 2622 | . . . . . . . 8 UnifSt UnifSt | |
6 | ressusp.2 | . . . . . . . 8 | |
7 | 4, 5, 6 | isusp 22065 | . . . . . . 7 UnifSp UnifSt UnifOn unifTopUnifSt |
8 | 3, 7 | sylib 208 | . . . . . 6 UnifSp UnifSt UnifOn unifTopUnifSt |
9 | 8 | simpld 475 | . . . . 5 UnifSp UnifSt UnifOn |
10 | simp2 1062 | . . . . . . 7 UnifSp | |
11 | 4, 6 | istps 20738 | . . . . . . 7 TopOn |
12 | 10, 11 | sylib 208 | . . . . . 6 UnifSp TopOn |
13 | simp3 1063 | . . . . . 6 UnifSp | |
14 | toponss 20731 | . . . . . 6 TopOn | |
15 | 12, 13, 14 | syl2anc 693 | . . . . 5 UnifSp |
16 | trust 22033 | . . . . 5 UnifSt UnifOn UnifSt ↾t UnifOn | |
17 | 9, 15, 16 | syl2anc 693 | . . . 4 UnifSp UnifSt ↾t UnifOn |
18 | 2, 17 | eqeltrd 2701 | . . 3 UnifSp UnifSt ↾s UnifOn |
19 | eqid 2622 | . . . . . 6 ↾s ↾s | |
20 | 19, 4 | ressbas2 15931 | . . . . 5 ↾s |
21 | 15, 20 | syl 17 | . . . 4 UnifSp ↾s |
22 | 21 | fveq2d 6195 | . . 3 UnifSp UnifOn UnifOn ↾s |
23 | 18, 22 | eleqtrd 2703 | . 2 UnifSp UnifSt ↾s UnifOn ↾s |
24 | 8 | simprd 479 | . . . . 5 UnifSp unifTopUnifSt |
25 | 13, 24 | eleqtrd 2703 | . . . 4 UnifSp unifTopUnifSt |
26 | restutopopn 22042 | . . . 4 UnifSt UnifOn unifTopUnifSt unifTopUnifSt ↾t unifTopUnifSt ↾t | |
27 | 9, 25, 26 | syl2anc 693 | . . 3 UnifSp unifTopUnifSt ↾t unifTopUnifSt ↾t |
28 | 24 | oveq1d 6665 | . . 3 UnifSp ↾t unifTopUnifSt ↾t |
29 | 2 | fveq2d 6195 | . . 3 UnifSp unifTopUnifSt ↾s unifTopUnifSt ↾t |
30 | 27, 28, 29 | 3eqtr4d 2666 | . 2 UnifSp ↾t unifTopUnifSt ↾s |
31 | eqid 2622 | . . 3 ↾s ↾s | |
32 | eqid 2622 | . . 3 UnifSt ↾s UnifSt ↾s | |
33 | 19, 6 | resstopn 20990 | . . 3 ↾t ↾s |
34 | 31, 32, 33 | isusp 22065 | . 2 ↾s UnifSp UnifSt ↾s UnifOn ↾s ↾t unifTopUnifSt ↾s |
35 | 23, 30, 34 | sylanbrc 698 | 1 UnifSp ↾s UnifSp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wss 3574 cxp 5112 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 ↾t crest 16081 ctopn 16082 TopOnctopon 20715 ctps 20736 UnifOncust 22003 unifTopcutop 22034 UnifStcuss 22057 UnifSpcusp 22058 |
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 |
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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 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-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-tset 15960 df-unif 15965 df-rest 16083 df-topn 16084 df-top 20699 df-topon 20716 df-topsp 20737 df-ust 22004 df-utop 22035 df-uss 22060 df-usp 22061 |
This theorem is referenced by: (None) |
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