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Mirrors > Home > MPE Home > Th. List > tsmspropd | Structured version Visualization version GIF version |
Description: The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 17930 etc. (Contributed by Mario Carneiro, 18-Sep-2015.) |
Ref | Expression |
---|---|
tsmspropd.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
tsmspropd.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
tsmspropd.h | ⊢ (𝜑 → 𝐻 ∈ 𝑋) |
tsmspropd.b | ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) |
tsmspropd.p | ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) |
tsmspropd.j | ⊢ (𝜑 → (TopOpen‘𝐺) = (TopOpen‘𝐻)) |
Ref | Expression |
---|---|
tsmspropd | ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmspropd.j | . . . 4 ⊢ (𝜑 → (TopOpen‘𝐺) = (TopOpen‘𝐻)) | |
2 | 1 | oveq1d 7165 | . . 3 ⊢ (𝜑 → ((TopOpen‘𝐺) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 ⊆ 𝑦}))) = ((TopOpen‘𝐻) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 ⊆ 𝑦})))) |
3 | tsmspropd.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
4 | resexg 5892 | . . . . . 6 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝑦) ∈ V) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝑦) ∈ V) |
6 | tsmspropd.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
7 | tsmspropd.h | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑋) | |
8 | tsmspropd.b | . . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) | |
9 | tsmspropd.p | . . . . 5 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) | |
10 | 5, 6, 7, 8, 9 | gsumpropd 17882 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦))) |
11 | 10 | mpteq2dv 5154 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦)))) |
12 | 2, 11 | fveq12d 6671 | . 2 ⊢ (𝜑 → (((TopOpen‘𝐺) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))) = (((TopOpen‘𝐻) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦))))) |
13 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
14 | eqid 2821 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
15 | eqid 2821 | . . 3 ⊢ (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin) | |
16 | eqid 2821 | . . 3 ⊢ ran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 ⊆ 𝑦}) = ran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 ⊆ 𝑦}) | |
17 | eqidd 2822 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹) | |
18 | 13, 14, 15, 16, 6, 3, 17 | tsmsval2 22732 | . 2 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (((TopOpen‘𝐺) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))))) |
19 | eqid 2821 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
20 | eqid 2821 | . . 3 ⊢ (TopOpen‘𝐻) = (TopOpen‘𝐻) | |
21 | 19, 20, 15, 16, 7, 3, 17 | tsmsval2 22732 | . 2 ⊢ (𝜑 → (𝐻 tsums 𝐹) = (((TopOpen‘𝐻) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦))))) |
22 | 12, 18, 21 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ↦ cmpt 5138 dom cdm 5549 ran crn 5550 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 Basecbs 16477 +gcplusg 16559 TopOpenctopn 16689 Σg cgsu 16708 filGencfg 20528 fLimf cflf 22537 tsums ctsu 22728 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-seq 13364 df-0g 16709 df-gsum 16710 df-tsms 22729 |
This theorem is referenced by: (None) |
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