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Mirrors > Home > MPE Home > Th. List > reldmpsr | Structured version Visualization version GIF version |
Description: The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
reldmpsr | β’ Rel dom mPwSer |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-psr 21235 | . 2 β’ mPwSer = (π β V, π β V β¦ β¦{β β (β0 βm π) β£ (β‘β β β) β Fin} / πβ¦β¦((Baseβπ) βm π) / πβ¦({β¨(Baseβndx), πβ©, β¨(+gβndx), ( βf (+gβπ) βΎ (π Γ π))β©, β¨(.rβndx), (π β π, π β π β¦ (π β π β¦ (π Ξ£g (π₯ β {π¦ β π β£ π¦ βr β€ π} β¦ ((πβπ₯)(.rβπ)(πβ(π βf β π₯)))))))β©} βͺ {β¨(Scalarβndx), πβ©, β¨( Β·π βndx), (π₯ β (Baseβπ), π β π β¦ ((π Γ {π₯}) βf (.rβπ)π))β©, β¨(TopSetβndx), (βtβ(π Γ {(TopOpenβπ)}))β©})) | |
2 | 1 | reldmmpo 7483 | 1 β’ Rel dom mPwSer |
Colors of variables: wff setvar class |
Syntax hints: β wcel 2107 {crab 3406 Vcvv 3444 β¦csb 3854 βͺ cun 3907 {csn 4585 {ctp 4589 β¨cop 4591 class class class wbr 5104 β¦ cmpt 5187 Γ cxp 5629 β‘ccnv 5630 dom cdm 5631 βΎ cres 5633 β cima 5634 Rel wrel 5636 βcfv 6492 (class class class)co 7350 β cmpo 7352 βf cof 7606 βr cofr 7607 βm cmap 8699 Fincfn 8817 β€ cle 11124 β cmin 11319 βcn 12087 β0cn0 12347 ndxcnx 17000 Basecbs 17018 +gcplusg 17068 .rcmulr 17069 Scalarcsca 17071 Β·π cvsca 17072 TopSetcts 17074 TopOpenctopn 17238 βtcpt 17255 Ξ£g cgsu 17257 mPwSer cmps 21230 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-xp 5637 df-rel 5638 df-dm 5641 df-oprab 7354 df-mpo 7355 df-psr 21235 |
This theorem is referenced by: psrbas 21270 psrelbas 21271 psrplusg 21273 psraddcl 21275 psrmulr 21276 psrmulcllem 21279 psrvscafval 21282 psrvscacl 21285 resspsrbas 21307 resspsradd 21308 resspsrmul 21309 mplval 21320 opsrle 21371 opsrbaslem 21373 opsrbaslemOLD 21374 psrbaspropd 21529 psropprmul 21532 |
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