Theorem tsmspropd 24165

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 18794 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 7453	. . 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 6053	. . . . 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 18713	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦)))
10	9	mpteq2dv 5253	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦))))
11	2, 10	fveq12d 6921	. 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 2737	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2737	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
14		eqid 2737	. . 3 ⊢ (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
15		eqid 2737	. . 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 2738	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
17	12, 13, 14, 15, 5, 3, 16	tsmsval2 24163	. 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 2737	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
19		eqid 2737	. . 3 ⊢ (TopOpen‘𝐻) = (TopOpen‘𝐻)
20	18, 19, 14, 15, 6, 3, 16	tsmsval2 24163	. 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 2787	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {crab 3436  Vcvv 3481   ∩ cin 3965   ⊆ wss 3966  𝒫 cpw 4608   ↦ cmpt 5234  dom cdm 5693  ran crn 5694   ↾ cres 5695  ‘cfv 6569  (class class class)co 7438  Fincfn 8993  Basecbs 17254  +_gcplusg 17307  TopOpenctopn 17477   Σ_g cgsu 17496  filGencfg 21380   fLimf cflf 23968   tsums ctsu 24159

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-pred 6329  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-frecs 8314  df-wrecs 8345  df-recs 8419  df-rdg 8458  df-seq 14049  df-0g 17497  df-gsum 17498  df-tsms 24160

This theorem is referenced by: (None)