Theorem tsmspropd 24001

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 18620 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 7355	. . 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 5973	. . . . 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 18539	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦)))
10	9	mpteq2dv 5182	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦))))
11	2, 10	fveq12d 6823	. 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 2729	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2729	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
14		eqid 2729	. . 3 ⊢ (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
15		eqid 2729	. . 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 2730	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
17	12, 13, 14, 15, 5, 3, 16	tsmsval2 23999	. 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 2729	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
19		eqid 2729	. . 3 ⊢ (TopOpen‘𝐻) = (TopOpen‘𝐻)
20	18, 19, 14, 15, 6, 3, 16	tsmsval2 23999	. 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 2774	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3392  Vcvv 3433   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4547   ↦ cmpt 5169  dom cdm 5613  ran crn 5614   ↾ cres 5615  ‘cfv 6476  (class class class)co 7340  Fincfn 8863  Basecbs 17107  +_gcplusg 17148  TopOpenctopn 17312   Σ_g cgsu 17331  filGencfg 21234   fLimf cflf 23804   tsums ctsu 23995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7343  df-oprab 7344  df-mpo 7345  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-seq 13897  df-0g 17332  df-gsum 17333  df-tsms 23996

This theorem is referenced by: (None)