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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 6672 | . . 3 ⊢ (𝑎 = 𝐴 → (ringLMod‘𝑎) = (ringLMod‘𝐴)) | |
2 | 1 | eleq1d 2899 | . 2 ⊢ (𝑎 = 𝐴 → ((ringLMod‘𝑎) ∈ LNoeM ↔ (ringLMod‘𝐴) ∈ LNoeM)) |
3 | df-lnr 39717 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Ringcrg 19299 ringLModcrglmod 19943 LNoeMclnm 39682 LNoeRclnr 39716 |
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 2795 |
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-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-lnr 39717 |
This theorem is referenced by: lnrring 39719 lnrlnm 39720 islnr2 39721 |
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