Theorem eldprd 19975

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 = (0g‘𝐺)
dprdval.w	⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }

Assertion

Ref	Expression
eldprd	⊢ (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))))

Distinct variable groups:   𝑓,ℎ,𝑖   𝐴,𝑓   𝑓,𝐼,ℎ,𝑖   𝑆,𝑓,ℎ,𝑖   𝑓,𝐺,ℎ,𝑖

Allowed substitution hints:   𝐴(ℎ,𝑖)   𝑊(𝑓,ℎ,𝑖)   0 (𝑓,ℎ,𝑖)

Proof of Theorem eldprd

Step	Hyp	Ref	Expression
1		elfvdm 6864	. . . . 5 ⊢ (𝐴 ∈ ( DProd ‘⟨𝐺, 𝑆⟩) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
2		df-ov 7362	. . . . 5 ⊢ (𝐺 DProd 𝑆) = ( DProd ‘⟨𝐺, 𝑆⟩)
3	1, 2	eleq2s 2859	. . . 4 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → ⟨𝐺, 𝑆⟩ ∈ dom DProd )
4		df-br 5075	. . . 4 ⊢ (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
5	3, 4	sylibr 236	. . 3 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆)
6	5	pm4.71ri 566	. 2 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)))
7		dprdval.0	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
8		dprdval.w	. . . . . . 7 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
9	7, 8	dprdval 19974	. . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))
10	9	eleq2d 2827	. . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))))
11		eqid 2741	. . . . . 6 ⊢ (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))
12		ovex 7392	. . . . . 6 ⊢ (𝐺 Σg 𝑓) ∈ V
13	11, 12	elrnmpti 5910	. . . . 5 ⊢ (𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))
14	10, 13	bitrdi 289	. . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))
15	14	ancoms 460	. . 3 ⊢ ((dom 𝑆 = 𝐼 ∧ 𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))
16	15	pm5.32da 585	. 2 ⊢ (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))))
17	6, 16	bitrid 285	1 ⊢ (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  {crab 3393  ⟨cop 4563   class class class wbr 5074   ↦ cmpt 5155  dom cdm 5620  ran crn 5621  ‘cfv 6488  (class class class)co 7359  Xcixp 8839   finSupp cfsupp 9268  0_gc0g 17397   Σ_g cgsu 17398   DProd cdprd 19964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-ixp 8840  df-dprd 19966

This theorem is referenced by:  dprdssv  19987  eldprdi  19989  dprdsubg  19995  dprdss  20000  dmdprdsplitlem  20008  dprddisj2  20010  dpjidcl  20029