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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 6901 | . . . 4 ⊢ 𝐵 ∈ V |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | fex2 7918 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
11 | 6, 7, 9, 10 | syl3anc 1372 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
12 | 6 | fdmd 6724 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
13 | 1, 2, 3, 4, 5, 11, 12 | tsmsval2 23615 | 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 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 ∩ cin 3945 ⊆ wss 3946 𝒫 cpw 4600 ↦ cmpt 5229 ran crn 5675 ↾ cres 5676 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 Fincfn 8934 Basecbs 17139 TopOpenctopn 17362 Σg cgsu 17381 filGencfg 20917 fLimf cflf 23420 tsums ctsu 23611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-fv 6547 df-ov 7406 df-oprab 7407 df-mpo 7408 df-tsms 23612 |
This theorem is referenced by: eltsms 23618 haustsms 23621 tsmscls 23623 tsmsmhm 23631 tsmsadd 23632 |
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