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Mirrors > Home > MPE Home > Th. List > ressxms | Structured version Visualization version Unicode version |
Description: The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
ressxms | ↾s |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . 6 | |
2 | eqid 2622 | . . . . . 6 | |
3 | 1, 2 | xmsxmet 22261 | . . . . 5 |
4 | 3 | adantr 481 | . . . 4 |
5 | xmetres 22169 | . . . 4 | |
6 | 4, 5 | syl 17 | . . 3 |
7 | resres 5409 | . . . . 5 | |
8 | inxp 5254 | . . . . . 6 | |
9 | 8 | reseq2i 5393 | . . . . 5 |
10 | 7, 9 | eqtri 2644 | . . . 4 |
11 | eqid 2622 | . . . . . . 7 ↾s ↾s | |
12 | eqid 2622 | . . . . . . 7 | |
13 | 11, 12 | ressds 16073 | . . . . . 6 ↾s |
14 | 13 | adantl 482 | . . . . 5 ↾s |
15 | incom 3805 | . . . . . . 7 | |
16 | 11, 1 | ressbas 15930 | . . . . . . . 8 ↾s |
17 | 16 | adantl 482 | . . . . . . 7 ↾s |
18 | 15, 17 | syl5eq 2668 | . . . . . 6 ↾s |
19 | 18 | sqxpeqd 5141 | . . . . 5 ↾s ↾s |
20 | 14, 19 | reseq12d 5397 | . . . 4 ↾s ↾s ↾s |
21 | 10, 20 | syl5eq 2668 | . . 3 ↾s ↾s ↾s |
22 | 18 | fveq2d 6195 | . . 3 ↾s |
23 | 6, 21, 22 | 3eltr3d 2715 | . 2 ↾s ↾s ↾s ↾s |
24 | eqid 2622 | . . . . . . 7 | |
25 | 24, 1, 2 | xmstopn 22256 | . . . . . 6 |
26 | 25 | adantr 481 | . . . . 5 |
27 | 26 | oveq1d 6665 | . . . 4 ↾t ↾t |
28 | inss1 3833 | . . . . 5 | |
29 | xpss12 5225 | . . . . . . . . 9 | |
30 | 28, 28, 29 | mp2an 708 | . . . . . . . 8 |
31 | resabs1 5427 | . . . . . . . 8 | |
32 | 30, 31 | ax-mp 5 | . . . . . . 7 |
33 | 10, 32 | eqtr4i 2647 | . . . . . 6 |
34 | eqid 2622 | . . . . . 6 | |
35 | eqid 2622 | . . . . . 6 | |
36 | 33, 34, 35 | metrest 22329 | . . . . 5 ↾t |
37 | 4, 28, 36 | sylancl 694 | . . . 4 ↾t |
38 | 27, 37 | eqtrd 2656 | . . 3 ↾t |
39 | xmstps 22258 | . . . . . . . . 9 | |
40 | 1, 24 | tpsuni 20740 | . . . . . . . . 9 |
41 | 39, 40 | syl 17 | . . . . . . . 8 |
42 | 41 | adantr 481 | . . . . . . 7 |
43 | 42 | ineq2d 3814 | . . . . . 6 |
44 | 15, 43 | syl5eq 2668 | . . . . 5 |
45 | 44 | oveq2d 6666 | . . . 4 ↾t ↾t |
46 | 1, 24 | istps 20738 | . . . . . 6 TopOn |
47 | 39, 46 | sylib 208 | . . . . 5 TopOn |
48 | eqid 2622 | . . . . . 6 | |
49 | 48 | restin 20970 | . . . . 5 TopOn ↾t ↾t |
50 | 47, 49 | sylan 488 | . . . 4 ↾t ↾t |
51 | 45, 50 | eqtr4d 2659 | . . 3 ↾t ↾t |
52 | 21 | fveq2d 6195 | . . 3 ↾s ↾s ↾s |
53 | 38, 51, 52 | 3eqtr3d 2664 | . 2 ↾t ↾s ↾s ↾s |
54 | 11, 24 | resstopn 20990 | . . 3 ↾t ↾s |
55 | eqid 2622 | . . 3 ↾s ↾s | |
56 | eqid 2622 | . . 3 ↾s ↾s ↾s ↾s ↾s ↾s | |
57 | 54, 55, 56 | isxms2 22253 | . 2 ↾s ↾s ↾s ↾s ↾s ↾t ↾s ↾s ↾s |
58 | 23, 53, 57 | sylanbrc 698 | 1 ↾s |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cin 3573 wss 3574 cuni 4436 cxp 5112 cres 5116 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cds 15950 ↾t crest 16081 ctopn 16082 cxmt 19731 cmopn 19736 TopOnctopon 20715 ctps 20736 cxme 22122 |
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-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 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-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-tset 15960 df-ds 15964 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-xms 22125 |
This theorem is referenced by: ressms 22331 qqhcn 30035 qqhucn 30036 |
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