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Theorem eldprd 20039
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 = (0g𝐺)
dprdval.w 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
Assertion
Ref Expression
eldprd (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))
Distinct variable groups:   𝑓,,𝑖   𝐴,𝑓   𝑓,𝐼,,𝑖   𝑆,𝑓,,𝑖   𝑓,𝐺,,𝑖
Allowed substitution hints:   𝐴(,𝑖)   𝑊(𝑓,,𝑖)   0 (𝑓,,𝑖)

Proof of Theorem eldprd
StepHypRef Expression
1 elfvdm 6944 . . . . 5 (𝐴 ∈ ( DProd ‘⟨𝐺, 𝑆⟩) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
2 df-ov 7434 . . . . 5 (𝐺 DProd 𝑆) = ( DProd ‘⟨𝐺, 𝑆⟩)
31, 2eleq2s 2857 . . . 4 (𝐴 ∈ (𝐺 DProd 𝑆) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
4 df-br 5149 . . . 4 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
53, 4sylibr 234 . . 3 (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆)
65pm4.71ri 560 . 2 (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆𝐴 ∈ (𝐺 DProd 𝑆)))
7 dprdval.0 . . . . . . 7 0 = (0g𝐺)
8 dprdval.w . . . . . . 7 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
97, 8dprdval 20038 . . . . . 6 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
109eleq2d 2825 . . . . 5 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓))))
11 eqid 2735 . . . . . 6 (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓))
12 ovex 7464 . . . . . 6 (𝐺 Σg 𝑓) ∈ V
1311, 12elrnmpti 5976 . . . . 5 (𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))
1410, 13bitrdi 287 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1514ancoms 458 . . 3 ((dom 𝑆 = 𝐼𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1615pm5.32da 579 . 2 (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))
176, 16bitrid 283 1 (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  cop 4637   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  cfv 6563  (class class class)co 7431  Xcixp 8936   finSupp cfsupp 9399  0gc0g 17486   Σg cgsu 17487   DProd cdprd 20028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-ixp 8937  df-dprd 20030
This theorem is referenced by:  dprdssv  20051  eldprdi  20053  dprdsubg  20059  dprdss  20064  dmdprdsplitlem  20072  dprddisj2  20074  dpjidcl  20093
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