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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 19110 | . 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 7264 | 1 ⊢ Rel dom DProd |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 {cab 2776 ∀wral 3106 {crab 3110 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 {csn 4525 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 “ cima 5522 Rel wrel 5524 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Xcixp 8444 finSupp cfsupp 8817 0gc0g 16705 Σg cgsu 16706 mrClscmrc 16846 Grpcgrp 18095 SubGrpcsubg 18265 Cntzccntz 18437 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-sep 5167 ax-nul 5174 ax-pr 5295 |
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-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-dm 5529 df-oprab 7139 df-mpo 7140 df-dprd 19110 |
This theorem is referenced by: dprddomprc 19115 dprdval0prc 19117 dprdval 19118 dprdgrp 19120 dprdf 19121 dprdssv 19131 subgdmdprd 19149 dprd2da 19157 dpjfval 19170 |
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