Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19177 | . 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 6669 | 1 ⊢ Rel dom mPwSer |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 {crab 2900 Vcvv 3173 ⦋csb 3499 ∪ cun 3538 {csn 4125 {ctp 4129 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ↾ cres 5040 “ cima 5041 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ∘𝑓 cof 6793 ∘𝑟 cofr 6794 ↑𝑚 cmap 7744 Fincfn 7841 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 TopSetcts 15774 TopOpenctopn 15905 ∏tcpt 15922 Σg cgsu 15924 mPwSer cmps 19172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-oprab 6553 df-mpt2 6554 df-psr 19177 |
This theorem is referenced by: psrbas 19199 psrelbas 19200 psrplusg 19202 psraddcl 19204 psrmulr 19205 psrmulcllem 19208 psrvscafval 19211 psrvscacl 19214 resspsrbas 19236 resspsradd 19237 resspsrmul 19238 mplval 19249 opsrle 19296 opsrbaslem 19298 opsrbaslemOLD 19299 psrbaspropd 19426 psropprmul 19429 |
Copyright terms: Public domain | W3C validator |