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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 | fvex 6113 | . . . . 5 ⊢ (Base‘𝐺) ∈ V | |
9 | 1, 8 | eqeltri 2684 | . . . 4 ⊢ 𝐵 ∈ V |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | fex2 7014 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
12 | 6, 7, 10, 11 | syl3anc 1318 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | fdm 5964 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
14 | 6, 13 | syl 17 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
15 | 1, 2, 3, 4, 5, 12, 14 | tsmsval2 21743 | 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 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 TopOpenctopn 15905 Σg cgsu 15924 filGencfg 19556 fLimf cflf 21549 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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-tsms 21740 |
This theorem is referenced by: eltsms 21746 haustsms 21749 tsmscls 21751 tsmsmhm 21759 tsmsadd 21760 |
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