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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 17928 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 7150 | . . 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 5864 | . . . . . 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 17880 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦))) |
11 | 10 | mpteq2dv 5126 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦)))) |
12 | 2, 11 | fveq12d 6652 | . 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 2798 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
14 | eqid 2798 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
15 | eqid 2798 | . . 3 ⊢ (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin) | |
16 | eqid 2798 | . . 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 2799 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹) | |
18 | 13, 14, 15, 16, 6, 3, 17 | tsmsval2 22735 | . 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 2798 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
20 | eqid 2798 | . . 3 ⊢ (TopOpen‘𝐻) = (TopOpen‘𝐻) | |
21 | 19, 20, 15, 16, 7, 3, 17 | tsmsval2 22735 | . 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 2843 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 Basecbs 16475 +gcplusg 16557 TopOpenctopn 16687 Σg cgsu 16706 filGencfg 20080 fLimf cflf 22540 tsums ctsu 22731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seq 13365 df-0g 16707 df-gsum 16708 df-tsms 22732 |
This theorem is referenced by: (None) |
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