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Mirrors > Home > MPE Home > Th. List > llyrest | Structured version Visualization version Unicode version |
Description: An open subspace of a locally space is also locally . (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
llyrest | Locally ↾t Locally |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | llytop 21275 | . . 3 Locally | |
2 | resttop 20964 | . . 3 ↾t | |
3 | 1, 2 | sylan 488 | . 2 Locally ↾t |
4 | restopn2 20981 | . . . . 5 ↾t | |
5 | 1, 4 | sylan 488 | . . . 4 Locally ↾t |
6 | simp1l 1085 | . . . . . . . . 9 Locally Locally | |
7 | simp2l 1087 | . . . . . . . . 9 Locally | |
8 | simp3 1063 | . . . . . . . . 9 Locally | |
9 | llyi 21277 | . . . . . . . . 9 Locally ↾t | |
10 | 6, 7, 8, 9 | syl3anc 1326 | . . . . . . . 8 Locally ↾t |
11 | simprl 794 | . . . . . . . . . . . . 13 Locally ↾t | |
12 | simprr1 1109 | . . . . . . . . . . . . . 14 Locally ↾t | |
13 | simpl2r 1115 | . . . . . . . . . . . . . 14 Locally ↾t | |
14 | 12, 13 | sstrd 3613 | . . . . . . . . . . . . 13 Locally ↾t |
15 | 6, 1 | syl 17 | . . . . . . . . . . . . . . 15 Locally |
16 | 15 | adantr 481 | . . . . . . . . . . . . . 14 Locally ↾t |
17 | simpl1r 1113 | . . . . . . . . . . . . . 14 Locally ↾t | |
18 | restopn2 20981 | . . . . . . . . . . . . . 14 ↾t | |
19 | 16, 17, 18 | syl2anc 693 | . . . . . . . . . . . . 13 Locally ↾t ↾t |
20 | 11, 14, 19 | mpbir2and 957 | . . . . . . . . . . . 12 Locally ↾t ↾t |
21 | selpw 4165 | . . . . . . . . . . . . 13 | |
22 | 12, 21 | sylibr 224 | . . . . . . . . . . . 12 Locally ↾t |
23 | 20, 22 | elind 3798 | . . . . . . . . . . 11 Locally ↾t ↾t |
24 | simprr2 1110 | . . . . . . . . . . 11 Locally ↾t | |
25 | restabs 20969 | . . . . . . . . . . . . 13 ↾t ↾t ↾t | |
26 | 16, 14, 17, 25 | syl3anc 1326 | . . . . . . . . . . . 12 Locally ↾t ↾t ↾t ↾t |
27 | simprr3 1111 | . . . . . . . . . . . 12 Locally ↾t ↾t | |
28 | 26, 27 | eqeltrd 2701 | . . . . . . . . . . 11 Locally ↾t ↾t ↾t |
29 | 23, 24, 28 | jca32 558 | . . . . . . . . . 10 Locally ↾t ↾t ↾t ↾t |
30 | 29 | ex 450 | . . . . . . . . 9 Locally ↾t ↾t ↾t ↾t |
31 | 30 | reximdv2 3014 | . . . . . . . 8 Locally ↾t ↾t ↾t ↾t |
32 | 10, 31 | mpd 15 | . . . . . . 7 Locally ↾t ↾t ↾t |
33 | 32 | 3expa 1265 | . . . . . 6 Locally ↾t ↾t ↾t |
34 | 33 | ralrimiva 2966 | . . . . 5 Locally ↾t ↾t ↾t |
35 | 34 | ex 450 | . . . 4 Locally ↾t ↾t ↾t |
36 | 5, 35 | sylbid 230 | . . 3 Locally ↾t ↾t ↾t ↾t |
37 | 36 | ralrimiv 2965 | . 2 Locally ↾t ↾t ↾t ↾t |
38 | islly 21271 | . 2 ↾t Locally ↾t ↾t ↾t ↾t ↾t | |
39 | 3, 37, 38 | sylanbrc 698 | 1 Locally ↾t Locally |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 cin 3573 wss 3574 cpw 4158 (class class class)co 6650 ↾t crest 16081 ctop 20698 Locally clly 21267 |
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-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-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-oadd 7564 df-er 7742 df-en 7956 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-lly 21269 |
This theorem is referenced by: loclly 21290 llyidm 21291 |
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