Theorem eldprd 19915

Description: A class 𝐴 is an internal direct product iff it is the (group) sum of an infinite, but finitely supported cartesian product of subgroups (which mutually commute and have trivial intersections). (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)

Hypotheses

Ref	Expression
dprdval.0	⊢ 0 = (0g‘𝐺)
dprdval.w	⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }

Assertion

Ref	Expression
eldprd	⊢ (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))))

Distinct variable groups:   𝑓,ℎ,𝑖   𝐴,𝑓   𝑓,𝐼,ℎ,𝑖   𝑆,𝑓,ℎ,𝑖   𝑓,𝐺,ℎ,𝑖

Allowed substitution hints:   𝐴(ℎ,𝑖)   𝑊(𝑓,ℎ,𝑖)   0 (𝑓,ℎ,𝑖)

Proof of Theorem eldprd

Step	Hyp	Ref	Expression
1		elfvdm 6927	. . . . 5 ⊢ (𝐴 ∈ ( DProd ‘⟨𝐺, 𝑆⟩) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
2		df-ov 7414	. . . . 5 ⊢ (𝐺 DProd 𝑆) = ( DProd ‘⟨𝐺, 𝑆⟩)
3	1, 2	eleq2s 2849	. . . 4 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
4		df-br 5148	. . . 4 ⊢ (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
5	3, 4	sylibr 233	. . 3 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆)
6	5	pm4.71ri 559	. 2 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)))
7		dprdval.0	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
8		dprdval.w	. . . . . . 7 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
9	7, 8	dprdval 19914	. . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))
10	9	eleq2d 2817	. . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))))
11		eqid 2730	. . . . . 6 ⊢ (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))
12		ovex 7444	. . . . . 6 ⊢ (𝐺 Σg 𝑓) ∈ V
13	11, 12	elrnmpti 5958	. . . . 5 ⊢ (𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))
14	10, 13	bitrdi 286	. . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))
15	14	ancoms 457	. . 3 ⊢ ((dom 𝑆 = 𝐼 ∧ 𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))
16	15	pm5.32da 577	. 2 ⊢ (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))))
17	6, 16	bitrid 282	1 ⊢ (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∃wrex 3068  {crab 3430  ⟨cop 4633   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676  ‘cfv 6542  (class class class)co 7411  Xcixp 8893   finSupp cfsupp 9363  0_gc0g 17389   Σ_g cgsu 17390   DProd cdprd 19904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-ixp 8894  df-dprd 19906

This theorem is referenced by:  dprdssv  19927  eldprdi  19929  dprdsubg  19935  dprdss  19940  dmdprdsplitlem  19948  dprddisj2  19950  dpjidcl  19969