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Mirrors > Home > MPE Home > Th. List > tsmscl | Structured version Visualization version GIF version |
Description: A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-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) |
Ref | Expression |
---|---|
tsmscl | ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmscl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2726 | . . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
3 | eqid 2726 | . . . 4 ⊢ (𝒫 𝐴 ∩ Fin) = (𝒫 𝐴 ∩ Fin) | |
4 | tsmscl.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | tsmscl.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TopSp) | |
6 | tsmscl.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | tsmscl.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | eltsms 24125 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑤 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑤 → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)(𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑤))))) |
9 | simpl 481 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ ∀𝑤 ∈ (TopOpen‘𝐺)(𝑥 ∈ 𝑤 → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)(𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑤))) → 𝑥 ∈ 𝐵) | |
10 | 8, 9 | biimtrdi 252 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ 𝐵)) |
11 | 10 | ssrdv 3984 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 ∩ cin 3945 ⊆ wss 3946 𝒫 cpw 4597 ↾ cres 5676 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 Fincfn 8966 Basecbs 17208 TopOpenctopn 17431 Σg cgsu 17450 CMndccmn 19774 TopSpctps 22922 tsums ctsu 24118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-seq 14016 df-hash 14343 df-0g 17451 df-gsum 17452 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-cntz 19307 df-cmn 19776 df-fbas 21336 df-fg 21337 df-top 22884 df-topon 22901 df-topsp 22923 df-ntr 23012 df-nei 23090 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-tsms 24119 |
This theorem is referenced by: tsmsmhm 24138 tsmsadd 24139 tsmssub 24141 tgptsmscls 24142 tgptsmscld 24143 taylfvallem 26382 esumcl 33876 |
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