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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 17139 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 6564 | . . 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 5362 | . . . . . 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 17095 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦))) |
11 | 10 | mpteq2dv 4673 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦)))) |
12 | 2, 11 | fveq12d 6109 | . 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 2610 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
14 | eqid 2610 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
15 | eqid 2610 | . . 3 ⊢ (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin) | |
16 | eqid 2610 | . . 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 2611 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹) | |
18 | 13, 14, 15, 16, 6, 3, 17 | tsmsval2 21743 | . 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 2610 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
20 | eqid 2610 | . . 3 ⊢ (TopOpen‘𝐻) = (TopOpen‘𝐻) | |
21 | 19, 20, 15, 16, 7, 3, 17 | tsmsval2 21743 | . 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 2654 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 +gcplusg 15768 TopOpenctopn 15905 Σg cgsu 15924 filGencfg 19556 fLimf cflf 21549 tsums ctsu 21739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-0g 15925 df-gsum 15926 df-tsms 21740 |
This theorem is referenced by: (None) |
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