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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 19598 | . 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 7408 | 1 ⊢ Rel dom DProd |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 {cab 2715 ∀wral 3064 {crab 3068 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 {csn 4561 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 “ cima 5592 Rel wrel 5594 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Xcixp 8685 finSupp cfsupp 9128 0gc0g 17150 Σg cgsu 17151 mrClscmrc 17292 Grpcgrp 18577 SubGrpcsubg 18749 Cntzccntz 18921 DProd cdprd 19596 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-dm 5599 df-oprab 7279 df-mpo 7280 df-dprd 19598 |
This theorem is referenced by: dprddomprc 19603 dprdval0prc 19605 dprdval 19606 dprdgrp 19608 dprdf 19609 dprdssv 19619 subgdmdprd 19637 dprd2da 19645 dpjfval 19658 |
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