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Mirrors > Home > MPE Home > Th. List > distgp | Structured version Visualization version Unicode version |
Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
distgp.1 | |
distgp.2 |
Ref | Expression |
---|---|
distgp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . 2 | |
2 | simpr 477 | . . . 4 | |
3 | distgp.1 | . . . . . 6 | |
4 | fvex 6201 | . . . . . 6 | |
5 | 3, 4 | eqeltri 2697 | . . . . 5 |
6 | distopon 20801 | . . . . 5 TopOn | |
7 | 5, 6 | ax-mp 5 | . . . 4 TopOn |
8 | 2, 7 | syl6eqel 2709 | . . 3 TopOn |
9 | distgp.2 | . . . 4 | |
10 | 3, 9 | istps 20738 | . . 3 TopOn |
11 | 8, 10 | sylibr 224 | . 2 |
12 | eqid 2622 | . . . . . 6 | |
13 | 3, 12 | grpsubf 17494 | . . . . 5 |
14 | 13 | adantr 481 | . . . 4 |
15 | 5, 5 | xpex 6962 | . . . . 5 |
16 | 5, 15 | elmap 7886 | . . . 4 |
17 | 14, 16 | sylibr 224 | . . 3 |
18 | 2, 2 | oveq12d 6668 | . . . . . 6 |
19 | txdis 21435 | . . . . . . 7 | |
20 | 5, 5, 19 | mp2an 708 | . . . . . 6 |
21 | 18, 20 | syl6eq 2672 | . . . . 5 |
22 | 21 | oveq1d 6665 | . . . 4 |
23 | cndis 21095 | . . . . 5 TopOn | |
24 | 15, 8, 23 | sylancr 695 | . . . 4 |
25 | 22, 24 | eqtrd 2656 | . . 3 |
26 | 17, 25 | eleqtrrd 2704 | . 2 |
27 | 9, 12 | istgp2 21895 | . 2 |
28 | 1, 11, 26, 27 | syl3anbrc 1246 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cpw 4158 cxp 5112 wf 5884 cfv 5888 (class class class)co 6650 cmap 7857 cbs 15857 ctopn 16082 cgrp 17422 csg 17424 TopOnctopon 20715 ctps 20736 ccn 21028 ctx 21363 ctgp 21875 |
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-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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-1st 7168 df-2nd 7169 df-map 7859 df-0g 16102 df-topgen 16104 df-plusf 17241 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-tx 21365 df-tmd 21876 df-tgp 21877 |
This theorem is referenced by: (None) |
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