Theorem tsmsval 22153

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		fvex 6342	. . . . 5 ⊢ (Base‘𝐺) ∈ V
9	1, 8	eqeltri 2845	. . . 4 ⊢ 𝐵 ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
11		fex2 7267	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
12	6, 7, 10, 11	syl3anc 1475	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
13		fdm 6191	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
14	6, 13	syl 17	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
15	1, 2, 3, 4, 5, 12, 14	tsmsval2 22152	1 ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))

Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144  {crab 3064  Vcvv 3349   ∩ cin 3720   ⊆ wss 3721  𝒫 cpw 4295   ↦ cmpt 4861  dom cdm 5249  ran crn 5250   ↾ cres 5251  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792  Fincfn 8108  Basecbs 16063  TopOpenctopn 16289   Σ_g cgsu 16308  filGencfg 19949   fLimf cflf 21958   tsums ctsu 22148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-tsms 22149

This theorem is referenced by:  eltsms  22155  haustsms  22158  tsmscls  22160  tsmsmhm  22168  tsmsadd  22169