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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 19857 | . 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 7538 | 1 ⊢ Rel dom DProd |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 {cab 2710 ∀wral 3062 {crab 3433 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 {csn 4627 ∪ cuni 4907 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 “ cima 5678 Rel wrel 5680 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 Xcixp 8887 finSupp cfsupp 9357 0gc0g 17381 Σg cgsu 17382 mrClscmrc 17523 Grpcgrp 18815 SubGrpcsubg 18994 Cntzccntz 19173 DProd cdprd 19855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-dm 5685 df-oprab 7408 df-mpo 7409 df-dprd 19857 |
This theorem is referenced by: dprddomprc 19862 dprdval0prc 19864 dprdval 19865 dprdgrp 19867 dprdf 19868 dprdssv 19878 subgdmdprd 19896 dprd2da 19904 dpjfval 19917 |
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