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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 766 | . 2 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π· β (CMetβπ)) | |
| 2 | simprl 770 | . 2 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π:ββΆ(Clsdβπ½)) | |
| 3 | cmetmet 25227 | . . . . . . . 8 β’ (π· β (CMetβπ) β π· β (Metβπ)) | |
| 4 | 3 | ad2antrr 725 | . . . . . . 7 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π· β (Metβπ)) |
| 5 | metxmet 24253 | . . . . . . 7 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
| 6 | bcth.2 | . . . . . . . 8 β’ π½ = (MetOpenβπ·) | |
| 7 | 6 | mopntopon 24358 | . . . . . . 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 22828 | . . . . . 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 772 | . . . . . 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 22841 | . . . . . . 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 2829 | . . . . 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 22999 | . . . . 5 β’ ((π½ β Top β§ βͺ ran π β π½) β ((intβπ½)ββͺ ran π) = βͺ ran π) | |
| 16 | 10, 14, 15 | syl2anc 583 | . . . 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 2768 | . . 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 768 | . . 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 3005 | . 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 25270 | . 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 1369 | 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 395 = wceq 1534 β wcel 2099 β wne 2937 βwrex 3067 β c0 4323 βͺ cuni 4908 ran crn 5679 βΆwf 6544 βcfv 6548 βcn 12243 βMetcxmet 21264 Metcmet 21265 MetOpencmopn 21269 Topctop 22808 TopOnctopon 22825 Clsdccld 22933 intcnt 22934 CMetccmet 25195 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-dc 10470 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ico 13363 df-rest 17404 df-topgen 17425 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-top 22809 df-topon 22826 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lm 23146 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-cfil 25196 df-cau 25197 df-cmet 25198 |
| This theorem is referenced by: ubthlem1 30693 |
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