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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 20286 | . 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 7285 | 1 ⊢ Rel dom evalSub |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 Vcvv 3494 ⦋csb 3883 {csn 4567 ↦ cmpt 5146 × cxp 5553 dom cdm 5555 ∘ ccom 5559 Rel wrel 5560 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 ↑m cmap 8406 Basecbs 16483 ↾s cress 16484 ↑s cpws 16720 CRingccrg 19298 RingHom crh 19464 SubRingcsubrg 19531 algSccascl 20084 mVar cmvr 20132 mPoly cmpl 20133 evalSub ces 20284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-dm 5565 df-oprab 7160 df-mpo 7161 df-evls 20286 |
This theorem is referenced by: mpfrcl 20298 evlval 20308 |
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