Theorem tsmsval 24160

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 6866	. . . 4 ⊢ 𝐵 ∈ V
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
10		fex2 7902	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
11	6, 7, 9, 10	syl3anc 1382	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
12	6	fdmd 6687	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
13	1, 2, 3, 4, 5, 11, 12	tsmsval2 24159	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   = wceq 1550   ∈ wcel 2132  {crab 3404  Vcvv 3444   ∩ cin 3894   ⊆ wss 3895  𝒫 cpw 4545   ↦ cmpt 5171  ran crn 5637   ↾ cres 5638  ⟶wf 6502  ‘cfv 6506  (class class class)co 7381  Fincfn 8912  Basecbs 17217  TopOpenctopn 17422   Σ_g cgsu 17441  filGencfg 21382   fLimf cflf 23964   tsums ctsu 24155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-tsms 24156

This theorem is referenced by:  eltsms  24162  haustsms  24165  tsmscls  24167  tsmsmhm  24175  tsmsadd  24176