Theorem tsmsval2 24256

Description: Definition of the topological group sum(s) of a collection 𝐹(𝑥) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
tsmsval.b	⊢ 𝐵 = (Base‘𝐺)
tsmsval.j	⊢ 𝐽 = (TopOpen‘𝐺)
tsmsval.s	⊢ 𝑆 = (𝒫 𝐴 ∩ Fin)
tsmsval.l	⊢ 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
tsmsval.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
tsmsval2.f	⊢ (𝜑 → 𝐹 ∈ 𝑊)
tsmsval2.a	⊢ (𝜑 → dom 𝐹 = 𝐴)

Assertion

Ref	Expression
tsmsval2	⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Distinct variable groups:   𝑦,𝑧,𝐹   𝑦,𝐺,𝑧   𝜑,𝑦,𝑧   𝑦,𝑆

Allowed substitution hints:   𝐴(𝑦,𝑧)   𝐵(𝑦,𝑧)   𝑆(𝑧)   𝐽(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑉(𝑦,𝑧)   𝑊(𝑦,𝑧)

Proof of Theorem tsmsval2

Dummy variables 𝑓 𝑠 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tsms 24253	. . 3 ⊢ tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋(𝒫 dom 𝑓 ∩ Fin) / 𝑠⦌(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦)))))
2	1	a1i 11	. 2 ⊢ (𝜑 → tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋(𝒫 dom 𝑓 ∩ Fin) / 𝑠⦌(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦))))))
3		vex 3467	. . . . . . 7 ⊢ 𝑓 ∈ V
4	3	dmex 7906	. . . . . 6 ⊢ dom 𝑓 ∈ V
5	4	pwex 5352	. . . . 5 ⊢ 𝒫 dom 𝑓 ∈ V
6	5	inex1 5288	. . . 4 ⊢ (𝒫 dom 𝑓 ∩ Fin) ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) ∈ V)
8		simplrl 788	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑤 = 𝐺)
9	8	fveq2d 6886	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (TopOpen‘𝑤) = (TopOpen‘𝐺))
10		tsmsval.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺)
11	9, 10	eqtr4di 2822	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (TopOpen‘𝑤) = 𝐽)
12		id 23	. . . . . . 7 ⊢ (𝑠 = (𝒫 dom 𝑓 ∩ Fin) → 𝑠 = (𝒫 dom 𝑓 ∩ Fin))
13		simprr 784	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
14	13	dmeqd 5896	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝑓 = dom 𝐹)
15		tsmsval2.a	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	15	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝐹 = 𝐴)
17	14, 16	eqtrd 2804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝑓 = 𝐴)
18	17	pweqd 4584	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝒫 dom 𝑓 = 𝒫 𝐴)
19	18	ineq1d 4180	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
20		tsmsval.s	. . . . . . . 8 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin)
21	19, 20	eqtr4di 2822	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
22	12, 21	sylan9eqr 2826	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑠 = 𝑆)
23	22	rabeqdv 3438	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
24	22, 23	mpteq12dv 5202	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}) = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
25	24	rneqd 5929	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}) = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
26		tsmsval.l	. . . . . . 7 ⊢ 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
27	25, 26	eqtr4di 2822	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}) = 𝐿)
28	22, 27	oveq12d 7429	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})) = (𝑆filGen𝐿))
29	11, 28	oveq12d 7429	. . . 4 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}))) = (𝐽 fLimf (𝑆filGen𝐿)))
30		simplrr 789	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑓 = 𝐹)
31	30	reseq1d 5978	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑓 ↾ 𝑦) = (𝐹 ↾ 𝑦))
32	8, 31	oveq12d 7429	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑤 Σg (𝑓 ↾ 𝑦)) = (𝐺 Σg (𝐹 ↾ 𝑦)))
33	22, 32	mpteq12dv 5202	. . . 4 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦))) = (𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦))))
34	29, 33	fveq12d 6889	. . 3 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦)))) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))
35	7, 34	csbied 3897	. 2 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → ⦋(𝒫 dom 𝑓 ∩ Fin) / 𝑠⦌(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦)))) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))
36		tsmsval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
37	36	elexd 3486	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
38		tsmsval2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
39	38	elexd 3486	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
40		fvexd 6897	. 2 ⊢ (𝜑 → ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))) ∈ V)
41	2, 35, 37, 39, 40	ovmpod 7563	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463  ⦋csb 3861   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567   ↦ cmpt 5196  dom cdm 5662  ran crn 5663   ↾ cres 5664  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  Fincfn 8943  Basecbs 17269  TopOpenctopn 17474   Σ_g cgsu 17493  filGencfg 21480   fLimf cflf 24061   tsums ctsu 24252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-tsms 24253

This theorem is referenced by:  tsmsval  24257  tsmspropd  24258