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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 6921 | . . . 4 ⊢ 𝐵 ∈ V |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | fex2 7957 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
11 | 6, 7, 9, 10 | syl3anc 1370 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
12 | 6 | fdmd 6747 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
13 | 1, 2, 3, 4, 5, 11, 12 | tsmsval2 24154 | 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 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ↦ cmpt 5231 ran crn 5690 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 Basecbs 17245 TopOpenctopn 17468 Σg cgsu 17487 filGencfg 21371 fLimf cflf 23959 tsums ctsu 24150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-tsms 24151 |
This theorem is referenced by: eltsms 24157 haustsms 24160 tsmscls 24162 tsmsmhm 24170 tsmsadd 24171 |
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