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| Mirrors > Home > MPE Home > Th. List > tgptsmscld | Structured version Visualization version GIF version | ||
| Description: The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| tgptsmscls.b | ⊢ 𝐵 = (Base‘𝐺) | 
| tgptsmscls.j | ⊢ 𝐽 = (TopOpen‘𝐺) | 
| tgptsmscls.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) | 
| tgptsmscls.2 | ⊢ (𝜑 → 𝐺 ∈ TopGrp) | 
| tgptsmscls.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| tgptsmscls.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | 
| Ref | Expression | 
|---|---|
| tgptsmscld | ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tgptsmscls.2 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TopGrp) | |
| 2 | tgptsmscls.j | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐺) | |
| 3 | tgptsmscls.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
| 4 | 2, 3 | tgptopon 24091 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵)) | 
| 5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵)) | 
| 6 | topontop 22920 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
| 7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) | 
| 8 | 0cld 23047 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) | |
| 9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → ∅ ∈ (Clsd‘𝐽)) | 
| 10 | eleq1 2828 | . . 3 ⊢ ((𝐺 tsums 𝐹) = ∅ → ((𝐺 tsums 𝐹) ∈ (Clsd‘𝐽) ↔ ∅ ∈ (Clsd‘𝐽))) | |
| 11 | 9, 10 | syl5ibrcom 247 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) = ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) | 
| 12 | n0 4352 | . . 3 ⊢ ((𝐺 tsums 𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) | |
| 13 | tgptsmscls.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
| 14 | 13 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd) | 
| 15 | 1 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp) | 
| 16 | tgptsmscls.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 17 | 16 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴 ∈ 𝑉) | 
| 18 | tgptsmscls.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 19 | 18 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴⟶𝐵) | 
| 20 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹)) | |
| 21 | 3, 2, 14, 15, 17, 19, 20 | tgptsmscls 24159 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑥})) | 
| 22 | tgptps 24089 | . . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
| 23 | 1, 22 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TopSp) | 
| 24 | 3, 13, 23, 16, 18 | tsmscl 24144 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) | 
| 25 | toponuni 22921 | . . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
| 26 | 5, 25 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽) | 
| 27 | 24, 26 | sseqtrd 4019 | . . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ∪ 𝐽) | 
| 28 | 27 | sselda 3982 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ∪ 𝐽) | 
| 29 | 28 | snssd 4808 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {𝑥} ⊆ ∪ 𝐽) | 
| 30 | eqid 2736 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 31 | 30 | clscld 23056 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) | 
| 32 | 7, 29, 31 | syl2an2r 685 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) | 
| 33 | 21, 32 | eqeltrd 2840 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) | 
| 34 | 33 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) | 
| 35 | 34 | exlimdv 1932 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) | 
| 36 | 12, 35 | biimtrid 242 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) ≠ ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) | 
| 37 | 11, 36 | pm2.61dne 3027 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 ⊆ wss 3950 ∅c0 4332 {csn 4625 ∪ cuni 4906 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 TopOpenctopn 17467 CMndccmn 19799 Topctop 22900 TopOnctopon 22917 TopSpctps 22939 Clsdccld 23025 clsccl 23027 TopGrpctgp 24080 tsums ctsu 24135 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-ec 8748 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-0g 17487 df-gsum 17488 df-topgen 17489 df-plusf 18653 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-eqg 19144 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-fbas 21362 df-fg 21363 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-cn 23236 df-cnp 23237 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-tmd 24081 df-tgp 24082 df-tsms 24136 | 
| This theorem is referenced by: (None) | 
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