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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 6103 | . . 3 ⊢ (𝑎 = 𝐴 → (ringLMod‘𝑎) = (ringLMod‘𝐴)) | |
2 | 1 | eleq1d 2672 | . 2 ⊢ (𝑎 = 𝐴 → ((ringLMod‘𝑎) ∈ LNoeM ↔ (ringLMod‘𝐴) ∈ LNoeM)) |
3 | df-lnr 36699 | . 2 ⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM} | |
4 | 2, 3 | elrab2 3333 | 1 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Ringcrg 18370 ringLModcrglmod 18990 LNoeMclnm 36663 LNoeRclnr 36698 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 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-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-lnr 36699 |
This theorem is referenced by: lnrring 36701 lnrlnm 36702 islnr2 36703 |
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