![]() |
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 ℂ. (Contributed by Mario Carneiro, 6-Jul-2017.) |
Ref | Expression |
---|---|
recld2.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
sszcld | ⊢ (𝐴 ⊆ ℤ → 𝐴 ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recld2.1 | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
2 | 1 | zcld2 24775 | . 2 ⊢ ℤ ∈ (Clsd‘𝐽) |
3 | id 22 | . . 3 ⊢ (𝐴 ⊆ ℤ → 𝐴 ⊆ ℤ) | |
4 | zex 12600 | . . . . 5 ⊢ ℤ ∈ V | |
5 | difss 4128 | . . . . 5 ⊢ (ℤ ∖ 𝐴) ⊆ ℤ | |
6 | 4, 5 | elpwi2 5349 | . . . 4 ⊢ (ℤ ∖ 𝐴) ∈ 𝒫 ℤ |
7 | 1 | zdis 24776 | . . . 4 ⊢ (𝐽 ↾t ℤ) = 𝒫 ℤ |
8 | 6, 7 | eleqtrri 2824 | . . 3 ⊢ (ℤ ∖ 𝐴) ∈ (𝐽 ↾t ℤ) |
9 | 1 | cnfldtopon 24743 | . . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
10 | zsscn 12599 | . . . . . 6 ⊢ ℤ ⊆ ℂ | |
11 | resttopon 23109 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ℤ ⊆ ℂ) → (𝐽 ↾t ℤ) ∈ (TopOn‘ℤ)) | |
12 | 9, 10, 11 | mp2an 690 | . . . . 5 ⊢ (𝐽 ↾t ℤ) ∈ (TopOn‘ℤ) |
13 | 12 | topontopi 22861 | . . . 4 ⊢ (𝐽 ↾t ℤ) ∈ Top |
14 | 12 | toponunii 22862 | . . . . 5 ⊢ ℤ = ∪ (𝐽 ↾t ℤ) |
15 | 14 | iscld 22975 | . . . 4 ⊢ ((𝐽 ↾t ℤ) ∈ Top → (𝐴 ∈ (Clsd‘(𝐽 ↾t ℤ)) ↔ (𝐴 ⊆ ℤ ∧ (ℤ ∖ 𝐴) ∈ (𝐽 ↾t ℤ)))) |
16 | 13, 15 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ (Clsd‘(𝐽 ↾t ℤ)) ↔ (𝐴 ⊆ ℤ ∧ (ℤ ∖ 𝐴) ∈ (𝐽 ↾t ℤ))) |
17 | 3, 8, 16 | sylanblrc 588 | . 2 ⊢ (𝐴 ⊆ ℤ → 𝐴 ∈ (Clsd‘(𝐽 ↾t ℤ))) |
18 | restcldr 23122 | . 2 ⊢ ((ℤ ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ (Clsd‘(𝐽 ↾t ℤ))) → 𝐴 ∈ (Clsd‘𝐽)) | |
19 | 2, 17, 18 | sylancr 585 | 1 ⊢ (𝐴 ⊆ ℤ → 𝐴 ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 𝒫 cpw 4604 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℤcz 12591 ↾t crest 17405 TopOpenctopn 17406 ℂfldccnfld 21296 Topctop 22839 TopOnctopon 22856 Clsdccld 22964 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 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 9436 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-fz 13520 df-fl 13793 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-rest 17407 df-topn 17408 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-xms 24270 df-ms 24271 |
This theorem is referenced by: lgamucov 27015 |
Copyright terms: Public domain | W3C validator |