Theorem tsmsval 24198

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)
tsmsval.a	⊢ (𝜑 → 𝐴 ∈ 𝑊)
tsmsval.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
tsmsval	⊢ (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 tsmsval

Step	Hyp	Ref	Expression
1		tsmsval.b	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		tsmsval.j	. 2 ⊢ 𝐽 = (TopOpen‘𝐺)
3		tsmsval.s	. 2 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin)
4		tsmsval.l	. 2 ⊢ 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})
5		tsmsval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6		tsmsval.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
7		tsmsval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
8	1	fvexi 6881	. . . 4 ⊢ 𝐵 ∈ V
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
10		fex2 7917	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
11	6, 7, 9, 10	syl3anc 1392	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
12	6	fdmd 6702	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
13	1, 2, 3, 4, 5, 11, 12	tsmsval2 24197	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  {crab 3415  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4556   ↦ cmpt 5182  ran crn 5649   ↾ cres 5650  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  Fincfn 8927  Basecbs 17255  TopOpenctopn 17460   Σ_g cgsu 17479  filGencfg 21420   fLimf cflf 24002   tsums ctsu 24193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-tsms 24194

This theorem is referenced by:  eltsms  24200  haustsms  24203  tsmscls  24205  tsmsmhm  24213  tsmsadd  24214