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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 17397 | . 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 7545 | 1 β’ Rel dom Xs |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 394 βwral 3059 Vcvv 3472 β¦csb 3892 βͺ cun 3945 β wss 3947 {csn 4627 {cpr 4629 {ctp 4631 β¨cop 4633 class class class wbr 5147 {copab 5209 β¦ cmpt 5230 Γ cxp 5673 dom cdm 5675 ran crn 5676 β ccom 5679 Rel wrel 5680 βcfv 6542 (class class class)co 7411 β cmpo 7413 1st c1st 7975 2nd c2nd 7976 Xcixp 8893 supcsup 9437 0cc0 11112 β*cxr 11251 < clt 11252 ndxcnx 17130 Basecbs 17148 +gcplusg 17201 .rcmulr 17202 Scalarcsca 17204 Β·π cvsca 17205 Β·πcip 17206 TopSetcts 17207 lecple 17208 distcds 17210 Hom chom 17212 compcco 17213 TopOpenctopn 17371 βtcpt 17388 Ξ£g cgsu 17390 Xscprds 17395 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-rab 3431 df-v 3474 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 7415 df-mpo 7416 df-prds 17397 |
This theorem is referenced by: dsmmval 21508 dsmmval2 21510 dsmmbas2 21511 dsmmfi 21512 |
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