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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 6942 | . . . . 5 ⊢ (𝐴 ∈ ( DProd ‘〈𝐺, 𝑆〉) → 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
| 2 | df-ov 7435 | . . . . 5 ⊢ (𝐺 DProd 𝑆) = ( DProd ‘〈𝐺, 𝑆〉) | |
| 3 | 1, 2 | eleq2s 2858 | . . . 4 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 〈𝐺, 𝑆〉 ∈ dom DProd ) | 
| 4 | df-br 5143 | . . . 4 ⊢ (𝐺dom DProd 𝑆 ↔ 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
| 5 | 3, 4 | sylibr 234 | . . 3 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆) | 
| 6 | 5 | pm4.71ri 560 | . 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 20024 | . . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))) | 
| 10 | 9 | eleq2d 2826 | . . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))) | 
| 11 | eqid 2736 | . . . . . 6 ⊢ (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) | |
| 12 | ovex 7465 | . . . . . 6 ⊢ (𝐺 Σg 𝑓) ∈ V | |
| 13 | 11, 12 | elrnmpti 5972 | . . . . 5 ⊢ (𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)) | 
| 14 | 10, 13 | bitrdi 287 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) | 
| 15 | 14 | ancoms 458 | . . 3 ⊢ ((dom 𝑆 = 𝐼 ∧ 𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) | 
| 16 | 15 | pm5.32da 579 | . 2 ⊢ (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) | 
| 17 | 6, 16 | bitrid 283 | 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 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 {crab 3435 〈cop 4631 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 ran crn 5685 ‘cfv 6560 (class class class)co 7432 Xcixp 8938 finSupp cfsupp 9402 0gc0g 17485 Σg cgsu 17486 DProd cdprd 20014 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-ixp 8939 df-dprd 20016 | 
| This theorem is referenced by: dprdssv 20037 eldprdi 20039 dprdsubg 20045 dprdss 20050 dmdprdsplitlem 20058 dprddisj2 20060 dpjidcl 20079 | 
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