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Theorem eldprd 19125
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 6701 . . . . 5 (𝐴 ∈ ( DProd ‘⟨𝐺, 𝑆⟩) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
2 df-ov 7158 . . . . 5 (𝐺 DProd 𝑆) = ( DProd ‘⟨𝐺, 𝑆⟩)
31, 2eleq2s 2931 . . . 4 (𝐴 ∈ (𝐺 DProd 𝑆) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
4 df-br 5066 . . . 4 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
53, 4sylibr 236 . . 3 (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆)
65pm4.71ri 563 . 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 19124 . . . . . 6 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
109eleq2d 2898 . . . . 5 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓))))
11 eqid 2821 . . . . . 6 (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓))
12 ovex 7188 . . . . . 6 (𝐺 Σg 𝑓) ∈ V
1311, 12elrnmpti 5831 . . . . 5 (𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))
1410, 13syl6bb 289 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1514ancoms 461 . . 3 ((dom 𝑆 = 𝐼𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1615pm5.32da 581 . 2 (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))
176, 16syl5bb 285 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 208  wa 398   = wceq 1533  wcel 2110  wrex 3139  {crab 3142  cop 4572   class class class wbr 5065  cmpt 5145  dom cdm 5554  ran crn 5555  cfv 6354  (class class class)co 7155  Xcixp 8460   finSupp cfsupp 8832  0gc0g 16712   Σg cgsu 16713   DProd cdprd 19114
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-ixp 8461  df-dprd 19116
This theorem is referenced by:  dprdssv  19137  eldprdi  19139  dprdsubg  19145  dprdss  19150  dmdprdsplitlem  19158  dprddisj2  19160  dpjidcl  19179
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