Theorem tsmsval2 22441

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 22438	. . 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 3419	. . . . . . 7 ⊢ 𝑓 ∈ V
4	3	dmex 7431	. . . . . 6 ⊢ dom 𝑓 ∈ V
5	4	pwex 5134	. . . . 5 ⊢ 𝒫 dom 𝑓 ∈ V
6	5	inex1 5078	. . . 4 ⊢ (𝒫 dom 𝑓 ∩ Fin) ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) ∈ V)
8		simplrl 764	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑤 = 𝐺)
9	8	fveq2d 6503	. . . . . 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	syl6eqr 2833	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (TopOpen‘𝑤) = 𝐽)
12		id 22	. . . . . . 7 ⊢ (𝑠 = (𝒫 dom 𝑓 ∩ Fin) → 𝑠 = (𝒫 dom 𝑓 ∩ Fin))
13		simprr 760	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
14	13	dmeqd 5624	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝑓 = dom 𝐹)
15		tsmsval2.a	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	15	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝐹 = 𝐴)
17	14, 16	eqtrd 2815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝑓 = 𝐴)
18	17	pweqd 4427	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝒫 dom 𝑓 = 𝒫 𝐴)
19	18	ineq1d 4076	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
20		tsmsval.s	. . . . . . . 8 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin)
21	19, 20	syl6eqr 2833	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
22	12, 21	sylan9eqr 2837	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑠 = 𝑆)
23	22	rabeqdv 3408	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
24	22, 23	mpteq12dv 5012	. . . . . . . 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 5651	. . . . . . 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	syl6eqr 2833	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}) = 𝐿)
28	22, 27	oveq12d 6994	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})) = (𝑆filGen𝐿))
29	11, 28	oveq12d 6994	. . . 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 765	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑓 = 𝐹)
31	30	reseq1d 5694	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑓 ↾ 𝑦) = (𝐹 ↾ 𝑦))
32	8, 31	oveq12d 6994	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑤 Σg (𝑓 ↾ 𝑦)) = (𝐺 Σg (𝐹 ↾ 𝑦)))
33	22, 32	mpteq12dv 5012	. . . 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 6506	. . 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 3816	. 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 3436	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
38		tsmsval2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
39	38	elexd 3436	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
40		fvexd 6514	. 2 ⊢ (𝜑 → ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))) ∈ V)
41	2, 35, 37, 39, 40	ovmpod 7118	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {crab 3093  Vcvv 3416  ⦋csb 3787   ∩ cin 3829   ⊆ wss 3830  𝒫 cpw 4422   ↦ cmpt 5008  dom cdm 5407  ran crn 5408   ↾ cres 5409  ‘cfv 6188  (class class class)co 6976   ∈ cmpo 6978  Fincfn 8306  Basecbs 16339  TopOpenctopn 16551   Σ_g cgsu 16570  filGencfg 20236   fLimf cflf 22247   tsums ctsu 22437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-tsms 22438

This theorem is referenced by:  tsmsval  22442  tsmspropd  22443