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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 2771 | . . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
3 | eqid 2771 | . . . 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 22156 | . . 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 468 | . . 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 | syl6bi 243 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ 𝐵)) |
11 | 10 | ssrdv 3758 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 ∩ cin 3722 ⊆ wss 3723 𝒫 cpw 4297 ↾ cres 5251 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 Fincfn 8109 Basecbs 16064 TopOpenctopn 16290 Σg cgsu 16309 CMndccmn 18400 TopSpctps 20957 tsums ctsu 22149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-fzo 12674 df-seq 13009 df-hash 13322 df-0g 16310 df-gsum 16311 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-cntz 17957 df-cmn 18402 df-fbas 19958 df-fg 19959 df-top 20919 df-topon 20936 df-topsp 20958 df-ntr 21045 df-nei 21123 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-tsms 22150 |
This theorem is referenced by: tsmsmhm 22169 tsmsadd 22170 tsmssub 22172 tgptsmscls 22173 tgptsmscld 22174 taylfvallem 24332 esumcl 30432 |
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