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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 17670 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 6921 | . . 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 5680 | . . . . . 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 17626 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦))) |
11 | 10 | mpteq2dv 4969 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦)))) |
12 | 2, 11 | fveq12d 6441 | . 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 2826 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
14 | eqid 2826 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
15 | eqid 2826 | . . 3 ⊢ (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin) | |
16 | eqid 2826 | . . 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 2827 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹) | |
18 | 13, 14, 15, 16, 6, 3, 17 | tsmsval2 22304 | . 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 2826 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
20 | eqid 2826 | . . 3 ⊢ (TopOpen‘𝐻) = (TopOpen‘𝐻) | |
21 | 19, 20, 15, 16, 7, 3, 17 | tsmsval2 22304 | . 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 2872 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 {crab 3122 Vcvv 3415 ∩ cin 3798 ⊆ wss 3799 𝒫 cpw 4379 ↦ cmpt 4953 dom cdm 5343 ran crn 5344 ↾ cres 5345 ‘cfv 6124 (class class class)co 6906 Fincfn 8223 Basecbs 16223 +gcplusg 16306 TopOpenctopn 16436 Σg cgsu 16455 filGencfg 20096 fLimf cflf 22110 tsums ctsu 22300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-seq 13097 df-0g 16456 df-gsum 16457 df-tsms 22301 |
This theorem is referenced by: (None) |
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