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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 764 | . 2 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π· β (CMetβπ)) | |
| 2 | simprl 768 | . 2 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π:ββΆ(Clsdβπ½)) | |
| 3 | cmetmet 25158 | . . . . . . . 8 β’ (π· β (CMetβπ) β π· β (Metβπ)) | |
| 4 | 3 | ad2antrr 723 | . . . . . . 7 β’ (((π· β (CMetβπ) β§ π β β ) β§ (π:ββΆ(Clsdβπ½) β§ βͺ ran π = π)) β π· β (Metβπ)) |
| 5 | metxmet 24184 | . . . . . . 7 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
| 6 | bcth.2 | . . . . . . . 8 β’ π½ = (MetOpenβπ·) | |
| 7 | 6 | mopntopon 24289 | . . . . . . 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 22759 | . . . . . 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 770 | . . . . . 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 22772 | . . . . . . 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 2825 | . . . . 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 22930 | . . . . 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 2764 | . . 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 766 | . . 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 3000 | . 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 25201 | . 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 1368 | 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 1533 β wcel 2098 β wne 2932 βwrex 3062 β c0 4315 βͺ cuni 4900 ran crn 5668 βΆwf 6530 βcfv 6534 βcn 12211 βMetcxmet 21219 Metcmet 21220 MetOpencmopn 21224 Topctop 22739 TopOnctopon 22756 Clsdccld 22864 intcnt 22865 CMetccmet 25126 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-dc 10438 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ico 13331 df-rest 17373 df-topgen 17394 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-top 22740 df-topon 22757 df-bases 22793 df-cld 22867 df-ntr 22868 df-cls 22869 df-nei 22946 df-lm 23077 df-fil 23694 df-fm 23786 df-flim 23787 df-flf 23788 df-cfil 25127 df-cau 25128 df-cmet 25129 |
| This theorem is referenced by: ubthlem1 30618 |
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