Theorem tsmspropd 24076

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 18684 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 7373	. . 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 5987	. . . . 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 18603	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐻 Σg (𝐹 ↾ 𝑦)))
10	9	mpteq2dv 5192	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹 ↾ 𝑦))))
11	2, 10	fveq12d 6841	. 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 24074	. 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 24074	. 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 2781	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554   ↦ cmpt 5179  dom cdm 5624  ran crn 5625   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358  Fincfn 8883  Basecbs 17136  +_gcplusg 17177  TopOpenctopn 17341   Σ_g cgsu 17360  filGencfg 21298   fLimf cflf 23879   tsums ctsu 24070

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seq 13925  df-0g 17361  df-gsum 17362  df-tsms 24071

This theorem is referenced by: (None)