Theorem tsmsval 22741

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 6686	. . . 4 ⊢ 𝐵 ∈ V
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
10		fex2 7640	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
11	6, 7, 9, 10	syl3anc 1367	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
12	6	fdmd 6525	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
13	1, 2, 3, 4, 5, 11, 12	tsmsval2 22740	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  {crab 3144  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541   ↦ cmpt 5148  ran crn 5558   ↾ cres 5559  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Fincfn 8511  Basecbs 16485  TopOpenctopn 16697   Σ_g cgsu 16716  filGencfg 20536   fLimf cflf 22545   tsums ctsu 22736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-tsms 22737

This theorem is referenced by:  eltsms  22743  haustsms  22746  tsmscls  22748  tsmsmhm  22756  tsmsadd  22757