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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 22693 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵)) |
6 | topontop 21524 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
8 | 0cld 21649 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → ∅ ∈ (Clsd‘𝐽)) |
10 | eleq1 2903 | . . 3 ⊢ ((𝐺 tsums 𝐹) = ∅ → ((𝐺 tsums 𝐹) ∈ (Clsd‘𝐽) ↔ ∅ ∈ (Clsd‘𝐽))) | |
11 | 9, 10 | syl5ibrcom 249 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) = ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
12 | n0 4313 | . . 3 ⊢ ((𝐺 tsums 𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) | |
13 | tgptsmscls.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
14 | 13 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd) |
15 | 1 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp) |
16 | tgptsmscls.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
17 | 16 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴 ∈ 𝑉) |
18 | tgptsmscls.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
19 | 18 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴⟶𝐵) |
20 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹)) | |
21 | 3, 2, 14, 15, 17, 19, 20 | tgptsmscls 22761 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑥})) |
22 | tgptps 22691 | . . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
23 | 1, 22 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
24 | 3, 13, 23, 16, 18 | tsmscl 22746 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
25 | toponuni 21525 | . . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
26 | 5, 25 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽) |
27 | 24, 26 | sseqtrd 4010 | . . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ∪ 𝐽) |
28 | 27 | sselda 3970 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ∪ 𝐽) |
29 | 28 | snssd 4745 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {𝑥} ⊆ ∪ 𝐽) |
30 | eqid 2824 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
31 | 30 | clscld 21658 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) |
32 | 7, 29, 31 | syl2an2r 683 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) |
33 | 21, 32 | eqeltrd 2916 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) |
34 | 33 | ex 415 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
35 | 34 | exlimdv 1933 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
36 | 12, 35 | syl5bi 244 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) ≠ ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
37 | 11, 36 | pm2.61dne 3106 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3019 ⊆ wss 3939 ∅c0 4294 {csn 4570 ∪ cuni 4841 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 TopOpenctopn 16698 CMndccmn 18909 Topctop 21504 TopOnctopon 21521 TopSpctps 21543 Clsdccld 21627 clsccl 21629 TopGrpctgp 22682 tsums ctsu 22737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-ec 8294 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-0g 16718 df-gsum 16719 df-topgen 16720 df-plusf 17854 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-eqg 18281 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-fbas 20545 df-fg 20546 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-cn 21838 df-cnp 21839 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-tmd 22683 df-tgp 22684 df-tsms 22738 |
This theorem is referenced by: (None) |
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