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| Mirrors > Home > MPE Home > Th. List > bcth2 | Structured version Visualization version GIF version | ||
| Description: Baire's Category Theorem, version 2: If countably many closed sets cover π, then one of them has an interior. (Contributed by Mario Carneiro, 10-Jan-2014.) |
| Ref | Expression |
|---|---|
| bcth.2 | β’ π½ = (MetOpenβπ·) |
| Ref | Expression |
|---|---|
| bcth2 | β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β βπ β β ((intβπ½)β(πβπ)) β β ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpll 765 | . 2 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π· β (CMetβπ)) | |
| 2 | simprl 769 | . 2 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π:ββΆ(Clsdβπ½)) | |
| 3 | cmetmet 24794 | . . . . . . . 8 β’ (π· β (CMetβπ) β π· β (Metβπ)) | |
| 4 | 3 | ad2antrr 724 | . . . . . . 7 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π· β (Metβπ)) |
| 5 | metxmet 23831 | . . . . . . 7 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
| 6 | bcth.2 | . . . . . . . 8 β’ π½ = (MetOpenβπ·) | |
| 7 | 6 | mopntopon 23936 | . . . . . . 7 β’ (π· β (βMetβπ) β π½ β (TopOnβπ)) |
| 8 | 4, 5, 7 | 3syl 18 | . . . . . 6 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π½ β (TopOnβπ)) |
| 9 | topontop 22406 | . . . . . 6 β’ (π½ β (TopOnβπ) β π½ β Top) | |
| 10 | 8, 9 | syl 17 | . . . . 5 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π½ β Top) |
| 11 | simprr 771 | . . . . . 6 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β βͺ ran π = π) | |
| 12 | toponmax 22419 | . . . . . . 7 β’ (π½ β (TopOnβπ) β π β π½) | |
| 13 | 8, 12 | syl 17 | . . . . . 6 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π β π½) |
| 14 | 11, 13 | eqeltrd 2833 | . . . . 5 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β βͺ ran π β π½) |
| 15 | isopn3i 22577 | . . . . 5 β’ ((π½ β Top β§ βͺ ran π β π½) β ((intβπ½)ββͺ ran π) = βͺ ran π) | |
| 16 | 10, 14, 15 | syl2anc 584 | . . . 4 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β ((intβπ½)ββͺ ran π) = βͺ ran π) |
| 17 | 16, 11 | eqtrd 2772 | . . 3 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β ((intβπ½)ββͺ ran π) = π) |
| 18 | simplr 767 | . . 3 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π β β ) | |
| 19 | 17, 18 | eqnetrd 3008 | . 2 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β ((intβπ½)ββͺ ran π) β β ) |
| 20 | 6 | bcth 24837 | . 2 β’ ((π· β (CMetβπ) β§ π:ββΆ(Clsdβπ½) β§ ((intβπ½)ββͺ ran π) β β ) β βπ β β ((intβπ½)β(πβπ)) β β ) |
| 21 | 1, 2, 19, 20 | syl3anc 1371 | 1 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β βπ β β ((intβπ½)β(πβπ)) β β ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 βwrex 3070 β c0 4321 βͺ cuni 4907 ran crn 5676 βΆwf 6536 βcfv 6540 βcn 12208 βMetcxmet 20921 Metcmet 20922 MetOpencmopn 20926 Topctop 22386 TopOnctopon 22403 Clsdccld 22511 intcnt 22512 CMetccmet 24762 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-dc 10437 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-rest 17364 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-top 22387 df-topon 22404 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lm 22724 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-cfil 24763 df-cau 24764 df-cmet 24765 |
| This theorem is referenced by: ubthlem1 30110 |
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