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Theorem eldprd 20004
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 6938 . . . . 5 (𝐴 ∈ ( DProd ‘⟨𝐺, 𝑆⟩) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
2 df-ov 7427 . . . . 5 (𝐺 DProd 𝑆) = ( DProd ‘⟨𝐺, 𝑆⟩)
31, 2eleq2s 2844 . . . 4 (𝐴 ∈ (𝐺 DProd 𝑆) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
4 df-br 5154 . . . 4 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
53, 4sylibr 233 . . 3 (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆)
65pm4.71ri 559 . 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 20003 . . . . . 6 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
109eleq2d 2812 . . . . 5 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓))))
11 eqid 2726 . . . . . 6 (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓))
12 ovex 7457 . . . . . 6 (𝐺 Σg 𝑓) ∈ V
1311, 12elrnmpti 5966 . . . . 5 (𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))
1410, 13bitrdi 286 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1514ancoms 457 . . 3 ((dom 𝑆 = 𝐼𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1615pm5.32da 577 . 2 (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))
176, 16bitrid 282 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 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  {crab 3419  cop 4639   class class class wbr 5153  cmpt 5236  dom cdm 5682  ran crn 5683  cfv 6554  (class class class)co 7424  Xcixp 8926   finSupp cfsupp 9405  0gc0g 17454   Σg cgsu 17455   DProd cdprd 19993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-ixp 8927  df-dprd 19995
This theorem is referenced by:  dprdssv  20016  eldprdi  20018  dprdsubg  20024  dprdss  20029  dmdprdsplitlem  20037  dprddisj2  20039  dpjidcl  20058
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