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Theorem eldprd 18610
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 6361 . . . . 5 (𝐴 ∈ ( DProd ‘⟨𝐺, 𝑆⟩) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
2 df-ov 6795 . . . . 5 (𝐺 DProd 𝑆) = ( DProd ‘⟨𝐺, 𝑆⟩)
31, 2eleq2s 2867 . . . 4 (𝐴 ∈ (𝐺 DProd 𝑆) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
4 df-br 4785 . . . 4 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
53, 4sylibr 224 . . 3 (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆)
65pm4.71ri 542 . 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 18609 . . . . . 6 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
109eleq2d 2835 . . . . 5 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓))))
11 eqid 2770 . . . . . 6 (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓))
12 ovex 6822 . . . . . 6 (𝐺 Σg 𝑓) ∈ V
1311, 12elrnmpti 5514 . . . . 5 (𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))
1410, 13syl6bb 276 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1514ancoms 455 . . 3 ((dom 𝑆 = 𝐼𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1615pm5.32da 560 . 2 (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))
176, 16syl5bb 272 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 196  wa 382   = wceq 1630  wcel 2144  wrex 3061  {crab 3064  cop 4320   class class class wbr 4784  cmpt 4861  dom cdm 5249  ran crn 5250  cfv 6031  (class class class)co 6792  Xcixp 8061   finSupp cfsupp 8430  0gc0g 16307   Σg cgsu 16308   DProd cdprd 18599
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-rep 4902  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-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  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-iun 4654  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-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-ixp 8062  df-dprd 18601
This theorem is referenced by:  dprdssv  18622  eldprdi  18624  dprdsubg  18630  dprdss  18635  dmdprdsplitlem  18643  dprddisj2  18645  dpjidcl  18664
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