MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldprd Structured version   Visualization version   GIF version

Theorem eldprd 18719
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 6443 . . . . 5 (𝐴 ∈ ( DProd ‘⟨𝐺, 𝑆⟩) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
2 df-ov 6881 . . . . 5 (𝐺 DProd 𝑆) = ( DProd ‘⟨𝐺, 𝑆⟩)
31, 2eleq2s 2896 . . . 4 (𝐴 ∈ (𝐺 DProd 𝑆) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
4 df-br 4844 . . . 4 (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
53, 4sylibr 226 . . 3 (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆)
65pm4.71ri 557 . 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 18718 . . . . . 6 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
109eleq2d 2864 . . . . 5 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓))))
11 eqid 2799 . . . . . 6 (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓))
12 ovex 6910 . . . . . 6 (𝐺 Σg 𝑓) ∈ V
1311, 12elrnmpti 5580 . . . . 5 (𝐴 ∈ ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))
1410, 13syl6bb 279 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1514ancoms 451 . . 3 ((dom 𝑆 = 𝐼𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓)))
1615pm5.32da 575 . 2 (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓𝑊 𝐴 = (𝐺 Σg 𝑓))))
176, 16syl5bb 275 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 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  {crab 3093  cop 4374   class class class wbr 4843  cmpt 4922  dom cdm 5312  ran crn 5313  cfv 6101  (class class class)co 6878  Xcixp 8148   finSupp cfsupp 8517  0gc0g 16415   Σg cgsu 16416   DProd cdprd 18708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-ixp 8149  df-dprd 18710
This theorem is referenced by:  dprdssv  18731  eldprdi  18733  dprdsubg  18739  dprdss  18744  dmdprdsplitlem  18752  dprddisj2  18754  dpjidcl  18773
  Copyright terms: Public domain W3C validator