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Mirrors > Home > MPE Home > Th. List > tsmsval | Structured version Visualization version GIF version |
Description: Definition of the topological group sum(s) of a collection 𝐹(𝑥) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tsmsval.b | ⊢ 𝐵 = (Base‘𝐺) |
tsmsval.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tsmsval.s | ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) |
tsmsval.l | ⊢ 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) |
tsmsval.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
tsmsval.a | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
tsmsval.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
tsmsval | ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmsval.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | tsmsval.j | . 2 ⊢ 𝐽 = (TopOpen‘𝐺) | |
3 | tsmsval.s | . 2 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) | |
4 | tsmsval.l | . 2 ⊢ 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) | |
5 | tsmsval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
6 | tsmsval.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
7 | tsmsval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
8 | 1 | fvexi 6709 | . . . 4 ⊢ 𝐵 ∈ V |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | fex2 7689 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
11 | 6, 7, 9, 10 | syl3anc 1373 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
12 | 6 | fdmd 6534 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
13 | 1, 2, 3, 4, 5, 11, 12 | tsmsval2 22981 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 {crab 3055 Vcvv 3398 ∩ cin 3852 ⊆ wss 3853 𝒫 cpw 4499 ↦ cmpt 5120 ran crn 5537 ↾ cres 5538 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 Basecbs 16666 TopOpenctopn 16880 Σg cgsu 16899 filGencfg 20306 fLimf cflf 22786 tsums ctsu 22977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-tsms 22978 |
This theorem is referenced by: eltsms 22984 haustsms 22987 tsmscls 22989 tsmsmhm 22997 tsmsadd 22998 |
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