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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 6702 | . . . . 5 ⊢ (𝐴 ∈ ( DProd ‘〈𝐺, 𝑆〉) → 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
2 | df-ov 7159 | . . . . 5 ⊢ (𝐺 DProd 𝑆) = ( DProd ‘〈𝐺, 𝑆〉) | |
3 | 1, 2 | eleq2s 2931 | . . . 4 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 〈𝐺, 𝑆〉 ∈ dom DProd ) |
4 | df-br 5067 | . . . 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 563 | . 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 19125 | . . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))) |
10 | 9 | eleq2d 2898 | . . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))) |
11 | eqid 2821 | . . . . . 6 ⊢ (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) | |
12 | ovex 7189 | . . . . . 6 ⊢ (𝐺 Σg 𝑓) ∈ V | |
13 | 11, 12 | elrnmpti 5832 | . . . . 5 ⊢ (𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)) |
14 | 10, 13 | syl6bb 289 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
15 | 14 | ancoms 461 | . . 3 ⊢ ((dom 𝑆 = 𝐼 ∧ 𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
16 | 15 | pm5.32da 581 | . 2 ⊢ (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) |
17 | 6, 16 | syl5bb 285 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {crab 3142 〈cop 4573 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ‘cfv 6355 (class class class)co 7156 Xcixp 8461 finSupp cfsupp 8833 0gc0g 16713 Σg cgsu 16714 DProd cdprd 19115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-ixp 8462 df-dprd 19117 |
This theorem is referenced by: dprdssv 19138 eldprdi 19140 dprdsubg 19146 dprdss 19151 dmdprdsplitlem 19159 dprddisj2 19161 dpjidcl 19180 |
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