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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 21696 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵)) |
6 | topontop 20541 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
8 | 0cld 20652 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → ∅ ∈ (Clsd‘𝐽)) |
10 | eleq1 2676 | . . 3 ⊢ ((𝐺 tsums 𝐹) = ∅ → ((𝐺 tsums 𝐹) ∈ (Clsd‘𝐽) ↔ ∅ ∈ (Clsd‘𝐽))) | |
11 | 9, 10 | syl5ibrcom 236 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) = ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
12 | n0 3890 | . . 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 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹)) | |
21 | 3, 2, 14, 15, 17, 19, 20 | tgptsmscls 21763 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑥})) |
22 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐽 ∈ Top) |
23 | tgptps 21694 | . . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
24 | 1, 23 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
25 | 3, 13, 24, 16, 18 | tsmscl 21748 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
26 | toponuni 20542 | . . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
27 | 5, 26 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽) |
28 | 25, 27 | sseqtrd 3604 | . . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ∪ 𝐽) |
29 | 28 | sselda 3568 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ∪ 𝐽) |
30 | 29 | snssd 4281 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {𝑥} ⊆ ∪ 𝐽) |
31 | eqid 2610 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
32 | 31 | clscld 20661 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) |
33 | 22, 30, 32 | syl2anc 691 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) |
34 | 21, 33 | eqeltrd 2688 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) |
35 | 34 | ex 449 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
36 | 35 | exlimdv 1848 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
37 | 12, 36 | syl5bi 231 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) ≠ ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
38 | 11, 37 | pm2.61dne 2868 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 {csn 4125 ∪ cuni 4372 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TopOpenctopn 15905 CMndccmn 18016 Topctop 20517 TopOnctopon 20518 TopSpctps 20519 Clsdccld 20630 clsccl 20632 TopGrpctgp 21685 tsums ctsu 21739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-topgen 15927 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-eqg 17416 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740 |
This theorem is referenced by: (None) |
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