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Mirrors > Home > MPE Home > Th. List > haustsms2 | Structured version Visualization version GIF version |
Description: In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 13-Sep-2015.) |
Ref | Expression |
---|---|
tsmscl.b | ⊢ 𝐵 = (Base‘𝐺) |
tsmscl.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
tsmscl.2 | ⊢ (𝜑 → 𝐺 ∈ TopSp) |
tsmscl.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
tsmscl.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
haustsms.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
haustsms.h | ⊢ (𝜑 → 𝐽 ∈ Haus) |
Ref | Expression |
---|---|
haustsms2 | ⊢ (𝜑 → (𝑋 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) = {𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ (𝐺 tsums 𝐹)) | |
2 | tsmscl.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺) | |
3 | tsmscl.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | tsmscl.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TopSp) | |
5 | tsmscl.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | tsmscl.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
7 | haustsms.j | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺) | |
8 | haustsms.h | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Haus) | |
9 | 2, 3, 4, 5, 6, 7, 8 | haustsms 22741 | . . . . . . 7 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) |
10 | 9 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐺 tsums 𝐹)) → ∃*𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) |
11 | eleq1 2877 | . . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ 𝑋 ∈ (𝐺 tsums 𝐹))) | |
12 | 11 | moi2 3655 | . . . . . . . 8 ⊢ (((𝑋 ∈ (𝐺 tsums 𝐹) ∧ ∃*𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ (𝑥 ∈ (𝐺 tsums 𝐹) ∧ 𝑋 ∈ (𝐺 tsums 𝐹))) → 𝑥 = 𝑋) |
13 | 12 | ancom2s 649 | . . . . . . 7 ⊢ (((𝑋 ∈ (𝐺 tsums 𝐹) ∧ ∃*𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ (𝑋 ∈ (𝐺 tsums 𝐹) ∧ 𝑥 ∈ (𝐺 tsums 𝐹))) → 𝑥 = 𝑋) |
14 | 13 | expr 460 | . . . . . 6 ⊢ (((𝑋 ∈ (𝐺 tsums 𝐹) ∧ ∃*𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑋 ∈ (𝐺 tsums 𝐹)) → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 = 𝑋)) |
15 | 1, 10, 1, 14 | syl21anc 836 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐺 tsums 𝐹)) → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 = 𝑋)) |
16 | velsn 4541 | . . . . 5 ⊢ (𝑥 ∈ {𝑋} ↔ 𝑥 = 𝑋) | |
17 | 15, 16 | syl6ibr 255 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐺 tsums 𝐹)) → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ {𝑋})) |
18 | 17 | ssrdv 3921 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) ⊆ {𝑋}) |
19 | snssi 4701 | . . . 4 ⊢ (𝑋 ∈ (𝐺 tsums 𝐹) → {𝑋} ⊆ (𝐺 tsums 𝐹)) | |
20 | 19 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐺 tsums 𝐹)) → {𝑋} ⊆ (𝐺 tsums 𝐹)) |
21 | 18, 20 | eqssd 3932 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) = {𝑋}) |
22 | 21 | ex 416 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) = {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃*wmo 2596 ⊆ wss 3881 {csn 4525 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 TopOpenctopn 16687 CMndccmn 18898 TopSpctps 21537 Hauscha 21913 tsums ctsu 22731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-cntz 18439 df-cmn 18900 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-topsp 21538 df-nei 21703 df-haus 21920 df-fil 22451 df-flim 22544 df-flf 22545 df-tsms 22732 |
This theorem is referenced by: haustsmsid 22746 xrge0tsms 23439 xrge0tsmsd 30742 |
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