![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sszcld | Structured version Visualization version GIF version |
Description: Every subset of the integers are closed in the topology on β. (Contributed by Mario Carneiro, 6-Jul-2017.) |
Ref | Expression |
---|---|
recld2.1 | β’ π½ = (TopOpenββfld) |
Ref | Expression |
---|---|
sszcld | β’ (π΄ β β€ β π΄ β (Clsdβπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recld2.1 | . . 3 β’ π½ = (TopOpenββfld) | |
2 | 1 | zcld2 24744 | . 2 β’ β€ β (Clsdβπ½) |
3 | id 22 | . . 3 β’ (π΄ β β€ β π΄ β β€) | |
4 | zex 12598 | . . . . 5 β’ β€ β V | |
5 | difss 4130 | . . . . 5 β’ (β€ β π΄) β β€ | |
6 | 4, 5 | elpwi2 5348 | . . . 4 β’ (β€ β π΄) β π« β€ |
7 | 1 | zdis 24745 | . . . 4 β’ (π½ βΎt β€) = π« β€ |
8 | 6, 7 | eleqtrri 2828 | . . 3 β’ (β€ β π΄) β (π½ βΎt β€) |
9 | 1 | cnfldtopon 24712 | . . . . . 6 β’ π½ β (TopOnββ) |
10 | zsscn 12597 | . . . . . 6 β’ β€ β β | |
11 | resttopon 23078 | . . . . . 6 β’ ((π½ β (TopOnββ) β§ β€ β β) β (π½ βΎt β€) β (TopOnββ€)) | |
12 | 9, 10, 11 | mp2an 691 | . . . . 5 β’ (π½ βΎt β€) β (TopOnββ€) |
13 | 12 | topontopi 22830 | . . . 4 β’ (π½ βΎt β€) β Top |
14 | 12 | toponunii 22831 | . . . . 5 β’ β€ = βͺ (π½ βΎt β€) |
15 | 14 | iscld 22944 | . . . 4 β’ ((π½ βΎt β€) β Top β (π΄ β (Clsdβ(π½ βΎt β€)) β (π΄ β β€ β§ (β€ β π΄) β (π½ βΎt β€)))) |
16 | 13, 15 | ax-mp 5 | . . 3 β’ (π΄ β (Clsdβ(π½ βΎt β€)) β (π΄ β β€ β§ (β€ β π΄) β (π½ βΎt β€))) |
17 | 3, 8, 16 | sylanblrc 589 | . 2 β’ (π΄ β β€ β π΄ β (Clsdβ(π½ βΎt β€))) |
18 | restcldr 23091 | . 2 β’ ((β€ β (Clsdβπ½) β§ π΄ β (Clsdβ(π½ βΎt β€))) β π΄ β (Clsdβπ½)) | |
19 | 2, 17, 18 | sylancr 586 | 1 β’ (π΄ β β€ β π΄ β (Clsdβπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 β cdif 3944 β wss 3947 π« cpw 4603 βcfv 6548 (class class class)co 7420 βcc 11137 β€cz 12589 βΎt crest 17402 TopOpenctopn 17403 βfldccnfld 21279 Topctop 22808 TopOnctopon 22825 Clsdccld 22933 |
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-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-tp 4634 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-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9435 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-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-fz 13518 df-fl 13790 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-rest 17404 df-topn 17405 df-topgen 17425 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-xms 24239 df-ms 24240 |
This theorem is referenced by: lgamucov 26983 |
Copyright terms: Public domain | W3C validator |