Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > reldmevls | Structured version Visualization version GIF version |
Description: Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
Ref | Expression |
---|---|
reldmevls | ⊢ Rel dom evalSub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-evls 20214 | . 2 ⊢ evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))))) | |
2 | 1 | reldmmpo 7274 | 1 ⊢ Rel dom evalSub |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 Vcvv 3492 ⦋csb 3880 {csn 4557 ↦ cmpt 5137 × cxp 5546 dom cdm 5548 ∘ ccom 5552 Rel wrel 5553 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 ↑m cmap 8395 Basecbs 16471 ↾s cress 16472 ↑s cpws 16708 CRingccrg 19227 RingHom crh 19393 SubRingcsubrg 19460 algSccascl 20012 mVar cmvr 20060 mPoly cmpl 20061 evalSub ces 20212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-dm 5558 df-oprab 7149 df-mpo 7150 df-evls 20214 |
This theorem is referenced by: mpfrcl 20226 evlval 20236 |
Copyright terms: Public domain | W3C validator |