Theorem tsmspropd 24086

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 18741 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypotheses

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))

Assertion

Ref	Expression
tsmspropd	⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))

Proof of Theorem tsmspropd

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsmspropd.j	. . . 4 ⊢ (𝜑 → (TopOpen‘𝐺) = (TopOpen‘𝐻))
2	1	oveq1d 7428	. . 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	3	resexd 6026	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝑦) ∈ V)
5		tsmspropd.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
6		tsmspropd.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
7		tsmspropd.b	. . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
8		tsmspropd.p	. . . . 5 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻))
9	4, 5, 6, 7, 8	gsumpropd 18660	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦)))
10	9	mpteq2dv 5224	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦))))
11	2, 10	fveq12d 6893	. 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)))))
12		eqid 2734	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2734	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
14		eqid 2734	. . 3 ⊢ (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
15		eqid 2734	. . 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})
16		eqidd 2735	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
17	12, 13, 14, 15, 5, 3, 16	tsmsval2 24084	. 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)))))
18		eqid 2734	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
19		eqid 2734	. . 3 ⊢ (TopOpen‘𝐻) = (TopOpen‘𝐻)
20	18, 19, 14, 15, 6, 3, 16	tsmsval2 24084	. 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)))))
21	11, 17, 20	3eqtr4d 2779	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  {crab 3419  Vcvv 3463   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580   ↦ cmpt 5205  dom cdm 5665  ran crn 5666   ↾ cres 5667  ‘cfv 6541  (class class class)co 7413  Fincfn 8967  Basecbs 17229  +_gcplusg 17273  TopOpenctopn 17437   Σ_g cgsu 17456  filGencfg 21315   fLimf cflf 23889   tsums ctsu 24080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-seq 14025  df-0g 17457  df-gsum 17458  df-tsms 24081

This theorem is referenced by: (None)