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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islnr | Structured version Visualization version GIF version |
Description: Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
islnr | ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6888 | . . 3 ⊢ (𝑎 = 𝐴 → (ringLMod‘𝑎) = (ringLMod‘𝐴)) | |
2 | 1 | eleq1d 2819 | . 2 ⊢ (𝑎 = 𝐴 → ((ringLMod‘𝑎) ∈ LNoeM ↔ (ringLMod‘𝐴) ∈ LNoeM)) |
3 | df-lnr 41785 | . 2 ⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM} | |
4 | 2, 3 | elrab2 3685 | 1 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6540 Ringcrg 20047 ringLModcrglmod 20770 LNoeMclnm 41750 LNoeRclnr 41784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-lnr 41785 |
This theorem is referenced by: lnrring 41787 lnrlnm 41788 islnr2 41789 |
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