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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 6677 | . . . . 5 ⊢ (𝐴 ∈ ( DProd ‘〈𝐺, 𝑆〉) → 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
2 | df-ov 7138 | . . . . 5 ⊢ (𝐺 DProd 𝑆) = ( DProd ‘〈𝐺, 𝑆〉) | |
3 | 1, 2 | eleq2s 2908 | . . . 4 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 〈𝐺, 𝑆〉 ∈ dom DProd ) |
4 | df-br 5031 | . . . 4 ⊢ (𝐺dom DProd 𝑆 ↔ 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
5 | 3, 4 | sylibr 237 | . . 3 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆) |
6 | 5 | pm4.71ri 564 | . 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 19118 | . . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))) |
10 | 9 | eleq2d 2875 | . . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))) |
11 | eqid 2798 | . . . . . 6 ⊢ (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) | |
12 | ovex 7168 | . . . . . 6 ⊢ (𝐺 Σg 𝑓) ∈ V | |
13 | 11, 12 | elrnmpti 5796 | . . . . 5 ⊢ (𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)) |
14 | 10, 13 | syl6bb 290 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
15 | 14 | ancoms 462 | . . 3 ⊢ ((dom 𝑆 = 𝐼 ∧ 𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
16 | 15 | pm5.32da 582 | . 2 ⊢ (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) |
17 | 6, 16 | syl5bb 286 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ‘cfv 6324 (class class class)co 7135 Xcixp 8444 finSupp cfsupp 8817 0gc0g 16705 Σg cgsu 16706 DProd cdprd 19108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-ixp 8445 df-dprd 19110 |
This theorem is referenced by: dprdssv 19131 eldprdi 19133 dprdsubg 19139 dprdss 19144 dmdprdsplitlem 19152 dprddisj2 19154 dpjidcl 19173 |
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