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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 | ⊢ Rel dom mPwSer |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-psr 21264 | . 2 ⊢ mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑m 𝑑) / 𝑏⦌({〈(Base‘ndx), 𝑏〉, 〈(+g‘ndx), ( ∘f (+g‘𝑟) ↾ (𝑏 × 𝑏))〉, 〈(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ∪ {〈(Scalar‘ndx), 𝑟〉, 〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r‘𝑟)𝑓))〉, 〈(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))〉})) | |
2 | 1 | reldmmpo 7484 | 1 ⊢ Rel dom mPwSer |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 {crab 3405 Vcvv 3443 ⦋csb 3853 ∪ cun 3906 {csn 4584 {ctp 4588 〈cop 4590 class class class wbr 5103 ↦ cmpt 5186 × cxp 5629 ◡ccnv 5630 dom cdm 5631 ↾ cres 5633 “ cima 5634 Rel wrel 5636 ‘cfv 6493 (class class class)co 7351 ∈ cmpo 7353 ∘f cof 7607 ∘r cofr 7608 ↑m cmap 8723 Fincfn 8841 ≤ cle 11148 − cmin 11343 ℕcn 12111 ℕ0cn0 12371 ndxcnx 17025 Basecbs 17043 +gcplusg 17093 .rcmulr 17094 Scalarcsca 17096 ·𝑠 cvsca 17097 TopSetcts 17099 TopOpenctopn 17263 ∏tcpt 17280 Σg cgsu 17282 mPwSer cmps 21259 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-dm 5641 df-oprab 7355 df-mpo 7356 df-psr 21264 |
This theorem is referenced by: psrbas 21299 psrelbas 21300 psrplusg 21302 psraddcl 21304 psrmulr 21305 psrmulcllem 21308 psrvscafval 21311 psrvscacl 21314 resspsrbas 21336 resspsradd 21337 resspsrmul 21338 mplval 21349 opsrle 21400 opsrbaslem 21402 opsrbaslemOLD 21403 psrbaspropd 21558 psropprmul 21561 |
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