Theorem tsmspropd 23389

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 18507 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 7352	. . 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 5970	. . . . 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 18459	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦)))
10	9	mpteq2dv 5194	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦))))
11	2, 10	fveq12d 6832	. 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 2736	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2736	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
14		eqid 2736	. . 3 ⊢ (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
15		eqid 2736	. . 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 2737	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
17	12, 13, 14, 15, 5, 3, 16	tsmsval2 23387	. 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 2736	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
19		eqid 2736	. . 3 ⊢ (TopOpen‘𝐻) = (TopOpen‘𝐻)
20	18, 19, 14, 15, 6, 3, 16	tsmsval2 23387	. 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 2786	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  {crab 3403  Vcvv 3441   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4547   ↦ cmpt 5175  dom cdm 5620  ran crn 5621   ↾ cres 5622  ‘cfv 6479  (class class class)co 7337  Fincfn 8804  Basecbs 17009  +_gcplusg 17059  TopOpenctopn 17229   Σ_g cgsu 17248  filGencfg 20692   fLimf cflf 23192   tsums ctsu 23383

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-seq 13823  df-0g 17249  df-gsum 17250  df-tsms 23384

This theorem is referenced by: (None)