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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 17936 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 7171 | . . 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 5898 | . . . . . 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 17888 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦))) |
11 | 10 | mpteq2dv 5162 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦)))) |
12 | 2, 11 | fveq12d 6677 | . 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 22738 | . 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 22738 | . 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 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 Basecbs 16483 +gcplusg 16565 TopOpenctopn 16695 Σg cgsu 16714 filGencfg 20534 fLimf cflf 22543 tsums ctsu 22734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seq 13371 df-0g 16715 df-gsum 16716 df-tsms 22735 |
This theorem is referenced by: (None) |
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