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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 6920 | . . . 4 ⊢ 𝐵 ∈ V |
| 9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
| 10 | fex2 7958 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
| 11 | 6, 7, 9, 10 | syl3anc 1373 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
| 12 | 6 | fdmd 6746 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
| 13 | 1, 2, 3, 4, 5, 11, 12 | tsmsval2 24138 | 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 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 Basecbs 17247 TopOpenctopn 17466 Σg cgsu 17485 filGencfg 21353 fLimf cflf 23943 tsums ctsu 24134 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-tsms 24135 |
| This theorem is referenced by: eltsms 24141 haustsms 24144 tsmscls 24146 tsmsmhm 24154 tsmsadd 24155 |
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