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Mirrors > Home > MPE Home > Th. List > fclsbas | Structured version Visualization version Unicode version |
Description: Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
fclsbas.f |
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Ref | Expression |
---|---|
fclsbas |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fclsbas.f |
. . . 4
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2 | fgcl 20971 |
. . . . 5
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3 | 2 | adantl 473 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 1, 3 | syl5eqel 2553 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | fclsopn 21107 |
. . 3
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6 | 4, 5 | syldan 478 |
. 2
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7 | ssfg 20965 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 7 | ad3antlr 745 |
. . . . . . . . . 10
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9 | 8, 1 | syl6sseqr 3465 |
. . . . . . . . 9
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10 | ssralv 3479 |
. . . . . . . . 9
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11 | 9, 10 | syl 17 |
. . . . . . . 8
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12 | ineq2 3619 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | neeq1d 2702 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 13 | cbvralv 3005 |
. . . . . . . 8
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15 | 11, 14 | syl6ib 234 |
. . . . . . 7
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16 | 1 | eleq2i 2541 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | elfg 20964 |
. . . . . . . . . . . 12
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18 | 17 | ad3antlr 745 |
. . . . . . . . . . 11
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19 | 16, 18 | syl5bb 265 |
. . . . . . . . . 10
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20 | 19 | simplbda 636 |
. . . . . . . . 9
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21 | r19.29r 2913 |
. . . . . . . . . . 11
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22 | sslin 3649 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | ssn0 3770 |
. . . . . . . . . . . . 13
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24 | 22, 23 | sylan 479 |
. . . . . . . . . . . 12
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25 | 24 | rexlimivw 2869 |
. . . . . . . . . . 11
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26 | 21, 25 | syl 17 |
. . . . . . . . . 10
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27 | 26 | ex 441 |
. . . . . . . . 9
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28 | 20, 27 | syl 17 |
. . . . . . . 8
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29 | 28 | ralrimdva 2812 |
. . . . . . 7
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30 | 15, 29 | impbid 195 |
. . . . . 6
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31 | 30 | anassrs 660 |
. . . . 5
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32 | 31 | pm5.74da 701 |
. . . 4
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33 | 32 | ralbidva 2828 |
. . 3
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34 | 33 | pm5.32da 653 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 6, 34 | bitrd 261 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-rep 4508 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-int 4227 df-iun 4271 df-iin 4272 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-fbas 19044 df-fg 19045 df-top 19998 df-topon 20000 df-cld 20111 df-ntr 20112 df-cls 20113 df-fil 20939 df-fcls 21034 |
This theorem is referenced by: (None) |
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