Theorem tsmsval2 24086

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 24083	. . 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 3446	. . . . . . 7 ⊢ 𝑓 ∈ V
4	3	dmex 7861	. . . . . 6 ⊢ dom 𝑓 ∈ V
5	4	pwex 5327	. . . . 5 ⊢ 𝒫 dom 𝑓 ∈ V
6	5	inex1 5264	. . . 4 ⊢ (𝒫 dom 𝑓 ∩ Fin) ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) ∈ V)
8		simplrl 777	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑤 = 𝐺)
9	8	fveq2d 6846	. . . . . 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 2790	. . . . 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 773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
14	13	dmeqd 5862	. . . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → dom 𝑓 = 𝐴)
18	17	pweqd 4573	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝒫 dom 𝑓 = 𝒫 𝐴)
19	18	ineq1d 4173	. . . . . . . 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 2790	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
22	12, 21	sylan9eqr 2794	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑠 = 𝑆)
23	22	rabeqdv 3416	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
24	22, 23	mpteq12dv 5187	. . . . . . . 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 5895	. . . . . . 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 2790	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦}) = 𝐿)
28	22, 27	oveq12d 7386	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})) = (𝑆filGen𝐿))
29	11, 28	oveq12d 7386	. . . 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 778	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑓 = 𝐹)
31	30	reseq1d 5945	. . . . . 6 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑓 ↾ 𝑦) = (𝐹 ↾ 𝑦))
32	8, 31	oveq12d 7386	. . . . 5 ⊢ (((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑤 Σg (𝑓 ↾ 𝑦)) = (𝐺 Σg (𝐹 ↾ 𝑦)))
33	22, 32	mpteq12dv 5187	. . . 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 6849	. . 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 3887	. 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 3466	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
38		tsmsval2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
39	38	elexd 3466	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
40		fvexd 6857	. 2 ⊢ (𝜑 → ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))) ∈ V)
41	2, 35, 37, 39, 40	ovmpod 7520	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442  ⦋csb 3851   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556   ↦ cmpt 5181  dom cdm 5632  ran crn 5633   ↾ cres 5634  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Fincfn 8895  Basecbs 17148  TopOpenctopn 17353   Σ_g cgsu 17372  filGencfg 21310   fLimf cflf 23891   tsums ctsu 24082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-tsms 24083

This theorem is referenced by:  tsmsval  24087  tsmspropd  24088