Theorem tsmsval2 21743

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 21740	. . 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 3176	. . . . . . 7 ⊢ 𝑓 ∈ V
4	3	dmex 6991	. . . . . 6 ⊢ dom 𝑓 ∈ V
5	4	pwex 4774	. . . . 5 ⊢ 𝒫 dom 𝑓 ∈ V
6	5	inex1 4727	. . . 4 ⊢ (𝒫 dom 𝑓 ∩ Fin) ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) ∈ V)
8		simplrl 796	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑤 = 𝐺)
9	8	fveq2d 6107	. . . . . 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 2662	. . . . 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 792	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
14	13	dmeqd 5248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝑓 = dom 𝐹)
15		tsmsval2.a	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	15	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝐹 = 𝐴)
17	14, 16	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝑓 = 𝐴)
18	17	pweqd 4113	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝒫 dom 𝑓 = 𝒫 𝐴)
19	18	ineq1d 3775	. . . . . . . 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 2662	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
22	12, 21	sylan9eqr 2666	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑠 = 𝑆)
23		rabeq 3166	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
24	22, 23	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
25	22, 24	mpteq12dv 4663	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}) = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}))
26	25	rneqd 5274	. . . . . . 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}))
27		tsmsval.l	. . . . . . 7 ⊢ 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
28	26, 27	syl6eqr 2662	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}) = 𝐿)
29	22, 28	oveq12d 6567	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})) = (𝑆filGen𝐿))
30	11, 29	oveq12d 6567	. . . 4 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}))) = (𝐽 fLimf (𝑆filGen𝐿)))
31		simplrr 797	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑓 = 𝐹)
32	31	reseq1d 5316	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑓 ↾ 𝑦) = (𝐹 ↾ 𝑦))
33	8, 32	oveq12d 6567	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑤 Σg (𝑓 ↾ 𝑦)) = (𝐺 Σg (𝐹 ↾ 𝑦)))
34	22, 33	mpteq12dv 4663	. . . 4 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦))) = (𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦))))
35	30, 34	fveq12d 6109	. . 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)))))
36	7, 35	csbied 3526	. 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)))))
37		tsmsval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
38		elex 3185	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
39	37, 38	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
40		tsmsval2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
41		elex 3185	. . 3 ⊢ (𝐹 ∈ 𝑊 → 𝐹 ∈ V)
42	40, 41	syl 17	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
43		fvex 6113	. . 3 ⊢ ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))) ∈ V
44	43	a1i 11	. 2 ⊢ (𝜑 → ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))) ∈ V)
45	2, 36, 39, 42, 44	ovmpt2d 6686	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⦋csb 3499   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  Basecbs 15695  TopOpenctopn 15905   Σ_g cgsu 15924  filGencfg 19556   fLimf cflf 21549   tsums ctsu 21739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-tsms 21740

This theorem is referenced by:  tsmsval  21744  tsmspropd  21745