Theorem tsmsval 24096

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 6855	. . . 4 ⊢ 𝐵 ∈ V
9	8	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
10		fex2 7887	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
11	6, 7, 9, 10	syl3anc 1374	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
12	6	fdmd 6679	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
13	1, 2, 3, 4, 5, 11, 12	tsmsval2 24095	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542   ↦ cmpt 5167  ran crn 5632   ↾ cres 5633  ⟶wf 6495  ‘cfv 6499  (class class class)co 7367  Fincfn 8893  Basecbs 17179  TopOpenctopn 17384   Σ_g cgsu 17403  filGencfg 21341   fLimf cflf 23900   tsums ctsu 24091

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 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

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 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-tsms 24092

This theorem is referenced by:  eltsms  24098  haustsms  24101  tsmscls  24103  tsmsmhm  24111  tsmsadd  24112