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Mirrors > Home > MPE Home > Th. List > eldprd | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dprdval.0 | ⊢ 0 = (0g‘𝐺) |
dprdval.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
Ref | Expression |
---|---|
eldprd | ⊢ (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6130 | . . . . 5 ⊢ (𝐴 ∈ ( DProd ‘〈𝐺, 𝑆〉) → 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
2 | df-ov 6552 | . . . . 5 ⊢ (𝐺 DProd 𝑆) = ( DProd ‘〈𝐺, 𝑆〉) | |
3 | 1, 2 | eleq2s 2706 | . . . 4 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 〈𝐺, 𝑆〉 ∈ dom DProd ) |
4 | df-br 4584 | . . . 4 ⊢ (𝐺dom DProd 𝑆 ↔ 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
5 | 3, 4 | sylibr 223 | . . 3 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆) |
6 | 5 | pm4.71ri 663 | . 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 18225 | . . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))) |
10 | 9 | eleq2d 2673 | . . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))) |
11 | eqid 2610 | . . . . . 6 ⊢ (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) | |
12 | ovex 6577 | . . . . . 6 ⊢ (𝐺 Σg 𝑓) ∈ V | |
13 | 11, 12 | elrnmpti 5297 | . . . . 5 ⊢ (𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)) |
14 | 10, 13 | syl6bb 275 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
15 | 14 | ancoms 468 | . . 3 ⊢ ((dom 𝑆 = 𝐼 ∧ 𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
16 | 15 | pm5.32da 671 | . 2 ⊢ (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) |
17 | 6, 16 | syl5bb 271 | 1 ⊢ (dom 𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 ‘cfv 5804 (class class class)co 6549 Xcixp 7794 finSupp cfsupp 8158 0gc0g 15923 Σg cgsu 15924 DProd cdprd 18215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-ixp 7795 df-dprd 18217 |
This theorem is referenced by: dprdssv 18238 eldprdi 18240 dprdsubg 18246 dprdss 18251 dmdprdsplitlem 18259 dprddisj2 18261 dpjidcl 18280 |
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