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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 6443 | . . . . 5 ⊢ (𝐴 ∈ ( DProd ‘〈𝐺, 𝑆〉) → 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
2 | df-ov 6881 | . . . . 5 ⊢ (𝐺 DProd 𝑆) = ( DProd ‘〈𝐺, 𝑆〉) | |
3 | 1, 2 | eleq2s 2896 | . . . 4 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 〈𝐺, 𝑆〉 ∈ dom DProd ) |
4 | df-br 4844 | . . . 4 ⊢ (𝐺dom DProd 𝑆 ↔ 〈𝐺, 𝑆〉 ∈ dom DProd ) | |
5 | 3, 4 | sylibr 226 | . . 3 ⊢ (𝐴 ∈ (𝐺 DProd 𝑆) → 𝐺dom DProd 𝑆) |
6 | 5 | pm4.71ri 557 | . 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 18718 | . . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))) |
10 | 9 | eleq2d 2864 | . . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ 𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))) |
11 | eqid 2799 | . . . . . 6 ⊢ (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) | |
12 | ovex 6910 | . . . . . 6 ⊢ (𝐺 Σg 𝑓) ∈ V | |
13 | 11, 12 | elrnmpti 5580 | . . . . 5 ⊢ (𝐴 ∈ ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)) |
14 | 10, 13 | syl6bb 279 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
15 | 14 | ancoms 451 | . . 3 ⊢ ((dom 𝑆 = 𝐼 ∧ 𝐺dom DProd 𝑆) → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
16 | 15 | pm5.32da 575 | . 2 ⊢ (dom 𝑆 = 𝐼 → ((𝐺dom DProd 𝑆 ∧ 𝐴 ∈ (𝐺 DProd 𝑆)) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) |
17 | 6, 16 | syl5bb 275 | 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 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 {crab 3093 〈cop 4374 class class class wbr 4843 ↦ cmpt 4922 dom cdm 5312 ran crn 5313 ‘cfv 6101 (class class class)co 6878 Xcixp 8148 finSupp cfsupp 8517 0gc0g 16415 Σg cgsu 16416 DProd cdprd 18708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-ixp 8149 df-dprd 18710 |
This theorem is referenced by: dprdssv 18731 eldprdi 18733 dprdsubg 18739 dprdss 18744 dmdprdsplitlem 18752 dprddisj2 18754 dpjidcl 18773 |
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