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Mirrors > Home > MPE Home > Th. List > reldmprds | Structured version Visualization version GIF version |
Description: The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.) |
Ref | Expression |
---|---|
reldmprds | β’ Rel dom Xs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-prds 17398 | . 2 β’ Xs = (π β V, π β V β¦ β¦Xπ₯ β dom π(Baseβ(πβπ₯)) / π£β¦β¦(π β π£, π β π£ β¦ Xπ₯ β dom π((πβπ₯)(Hom β(πβπ₯))(πβπ₯))) / ββ¦(({β¨(Baseβndx), π£β©, β¨(+gβndx), (π β π£, π β π£ β¦ (π₯ β dom π β¦ ((πβπ₯)(+gβ(πβπ₯))(πβπ₯))))β©, β¨(.rβndx), (π β π£, π β π£ β¦ (π₯ β dom π β¦ ((πβπ₯)(.rβ(πβπ₯))(πβπ₯))))β©} βͺ {β¨(Scalarβndx), π β©, β¨( Β·π βndx), (π β (Baseβπ ), π β π£ β¦ (π₯ β dom π β¦ (π( Β·π β(πβπ₯))(πβπ₯))))β©, β¨(Β·πβndx), (π β π£, π β π£ β¦ (π Ξ£g (π₯ β dom π β¦ ((πβπ₯)(Β·πβ(πβπ₯))(πβπ₯)))))β©}) βͺ ({β¨(TopSetβndx), (βtβ(TopOpen β π))β©, β¨(leβndx), {β¨π, πβ© β£ ({π, π} β π£ β§ βπ₯ β dom π(πβπ₯)(leβ(πβπ₯))(πβπ₯))}β©, β¨(distβndx), (π β π£, π β π£ β¦ sup((ran (π₯ β dom π β¦ ((πβπ₯)(distβ(πβπ₯))(πβπ₯))) βͺ {0}), β*, < ))β©} βͺ {β¨(Hom βndx), ββ©, β¨(compβndx), (π β (π£ Γ π£), π β π£ β¦ (π β ((2nd βπ)βπ), π β (ββπ) β¦ (π₯ β dom π β¦ ((πβπ₯)(β¨((1st βπ)βπ₯), ((2nd βπ)βπ₯)β©(compβ(πβπ₯))(πβπ₯))(πβπ₯)))))β©}))) | |
2 | 1 | reldmmpo 7546 | 1 β’ Rel dom Xs |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 βwral 3060 Vcvv 3473 β¦csb 3893 βͺ cun 3946 β wss 3948 {csn 4628 {cpr 4630 {ctp 4632 β¨cop 4634 class class class wbr 5148 {copab 5210 β¦ cmpt 5231 Γ cxp 5674 dom cdm 5676 ran crn 5677 β ccom 5680 Rel wrel 5681 βcfv 6543 (class class class)co 7412 β cmpo 7414 1st c1st 7977 2nd c2nd 7978 Xcixp 8895 supcsup 9439 0cc0 11114 β*cxr 11252 < clt 11253 ndxcnx 17131 Basecbs 17149 +gcplusg 17202 .rcmulr 17203 Scalarcsca 17205 Β·π cvsca 17206 Β·πcip 17207 TopSetcts 17208 lecple 17209 distcds 17211 Hom chom 17213 compcco 17214 TopOpenctopn 17372 βtcpt 17389 Ξ£g cgsu 17391 Xscprds 17396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-dm 5686 df-oprab 7416 df-mpo 7417 df-prds 17398 |
This theorem is referenced by: dsmmval 21509 dsmmval2 21511 dsmmbas2 21512 dsmmfi 21513 |
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