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| Mirrors > Home > MPE Home > Th. List > reldmdprd | Structured version Visualization version GIF version | ||
| Description: The domain of the internal direct product operation is a relation. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.) |
| Ref | Expression |
|---|---|
| reldmdprd | ⊢ Rel dom DProd |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-dprd 20015 | . 2 ⊢ DProd = (𝑔 ∈ Grp, 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓))) | |
| 2 | 1 | reldmmpo 7567 | 1 ⊢ Rel dom DProd |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 {cab 2714 ∀wral 3061 {crab 3436 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 “ cima 5688 Rel wrel 5690 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Xcixp 8937 finSupp cfsupp 9401 0gc0g 17484 Σg cgsu 17485 mrClscmrc 17626 Grpcgrp 18951 SubGrpcsubg 19138 Cntzccntz 19333 DProd cdprd 20013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-oprab 7435 df-mpo 7436 df-dprd 20015 |
| This theorem is referenced by: dprddomprc 20020 dprdval0prc 20022 dprdval 20023 dprdgrp 20025 dprdf 20026 dprdssv 20036 subgdmdprd 20054 dprd2da 20062 dpjfval 20075 |
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