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Mirrors > Home > MPE Home > Th. List > nrmr0reg | Structured version Visualization version Unicode version |
Description: A normal R0 space is also regular. These spaces are usually referred to as normal regular spaces. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
nrmr0reg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrmtop 20401 |
. . 3
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2 | 1 | adantr 471 |
. 2
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3 | simpll 765 |
. . . . 5
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4 | simprl 769 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 2 | adantr 471 |
. . . . . . 7
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6 | eqid 2462 |
. . . . . . . 8
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7 | 6 | toptopon 19997 |
. . . . . . 7
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8 | 5, 7 | sylib 201 |
. . . . . 6
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9 | simplr 767 |
. . . . . 6
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10 | simprr 771 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | elunii 4217 |
. . . . . . 7
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12 | 10, 4, 11 | syl2anc 671 |
. . . . . 6
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13 | eqid 2462 |
. . . . . . 7
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14 | 13 | r0cld 20802 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 8, 9, 12, 14 | syl3anc 1276 |
. . . . 5
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16 | simp1rr 1080 |
. . . . . . 7
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17 | 4 | adantr 471 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | elequ2 1912 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | elequ2 1912 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 18, 19 | bibi12d 327 |
. . . . . . . . . 10
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21 | 20 | rspcv 3158 |
. . . . . . . . 9
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22 | 17, 21 | syl 17 |
. . . . . . . 8
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23 | 22 | 3impia 1212 |
. . . . . . 7
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24 | 16, 23 | mpbird 240 |
. . . . . 6
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25 | 24 | rabssdv 3521 |
. . . . 5
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26 | nrmsep3 20420 |
. . . . 5
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27 | 3, 4, 15, 25, 26 | syl13anc 1278 |
. . . 4
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28 | biidd 245 |
. . . . . . . . 9
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29 | 28 | ralrimivw 2815 |
. . . . . . . 8
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30 | elequ1 1905 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 30 | bibi1d 325 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 31 | ralbidv 2839 |
. . . . . . . . 9
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33 | 32 | elrab 3208 |
. . . . . . . 8
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34 | 12, 29, 33 | sylanbrc 675 |
. . . . . . 7
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35 | ssel 3438 |
. . . . . . 7
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36 | 34, 35 | syl5com 31 |
. . . . . 6
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37 | 36 | anim1d 572 |
. . . . 5
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38 | 37 | reximdv 2873 |
. . . 4
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39 | 27, 38 | mpd 15 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 39 | ralrimivva 2821 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | isreg 20397 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
42 | 2, 40, 41 | sylanbrc 675 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-map 7500 df-qtop 15455 df-top 19970 df-topon 19972 df-cld 20083 df-cn 20292 df-t1 20379 df-reg 20381 df-nrm 20382 df-kq 20758 |
This theorem is referenced by: nrmreg 20888 |
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