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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 19356 | . 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 | reldmmpt2 6771 | 1 ⊢ Rel dom mPwSer |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 {crab 2916 Vcvv 3200 ⦋csb 3533 ∪ cun 3572 {csn 4177 {ctp 4181 〈cop 4183 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 dom cdm 5114 ↾ cres 5116 “ cima 5117 Rel wrel 5119 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ∘𝑓 cof 6895 ∘𝑟 cofr 6896 ↑𝑚 cmap 7857 Fincfn 7955 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ0cn0 11292 ndxcnx 15854 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 TopSetcts 15947 TopOpenctopn 16082 ∏tcpt 16099 Σg cgsu 16101 mPwSer cmps 19351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-dm 5124 df-oprab 6654 df-mpt2 6655 df-psr 19356 |
This theorem is referenced by: psrbas 19378 psrelbas 19379 psrplusg 19381 psraddcl 19383 psrmulr 19384 psrmulcllem 19387 psrvscafval 19390 psrvscacl 19393 resspsrbas 19415 resspsradd 19416 resspsrmul 19417 mplval 19428 opsrle 19475 opsrbaslem 19477 opsrbaslemOLD 19478 psrbaspropd 19605 psropprmul 19608 |
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