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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prmidlssidl | Structured version Visualization version GIF version |
Description: Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
Ref | Expression |
---|---|
prmidlssidl | ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmidlidl 31125 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) → 𝑖 ∈ (LIdeal‘𝑅)) | |
2 | 1 | ex 417 | . 2 ⊢ (𝑅 ∈ Ring → (𝑖 ∈ (PrmIdeal‘𝑅) → 𝑖 ∈ (LIdeal‘𝑅))) |
3 | 2 | ssrdv 3894 | 1 ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ⊆ wss 3854 ‘cfv 6328 Ringcrg 19350 LIdealclidl 19995 PrmIdealcprmidl 31116 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-iota 6287 df-fun 6330 df-fv 6336 df-ov 7146 df-prmidl 31117 |
This theorem is referenced by: 0ringprmidl 31131 rspecbas 31321 rspectopn 31323 |
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