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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prmidlnr | Structured version Visualization version GIF version |
Description: A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.) |
Ref | Expression |
---|---|
prmidlval.1 | β’ π΅ = (Baseβπ ) |
prmidlval.2 | β’ Β· = (.rβπ ) |
Ref | Expression |
---|---|
prmidlnr | β’ ((π β Ring β§ π β (PrmIdealβπ )) β π β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmidlval.1 | . . . 4 β’ π΅ = (Baseβπ ) | |
2 | prmidlval.2 | . . . 4 β’ Β· = (.rβπ ) | |
3 | 1, 2 | isprmidl 33187 | . . 3 β’ (π β Ring β (π β (PrmIdealβπ ) β (π β (LIdealβπ ) β§ π β π΅ β§ βπ β (LIdealβπ )βπ β (LIdealβπ )(βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π β (π β π β¨ π β π))))) |
4 | 3 | biimpa 475 | . 2 β’ ((π β Ring β§ π β (PrmIdealβπ )) β (π β (LIdealβπ ) β§ π β π΅ β§ βπ β (LIdealβπ )βπ β (LIdealβπ )(βπ₯ β π βπ¦ β π (π₯ Β· π¦) β π β (π β π β¨ π β π)))) |
5 | 4 | simp2d 1140 | 1 β’ ((π β Ring β§ π β (PrmIdealβπ )) β π β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β¨ wo 845 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 βwral 3058 β wss 3949 βcfv 6553 (class class class)co 7426 Basecbs 17189 .rcmulr 17243 Ringcrg 20187 LIdealclidl 21116 PrmIdealcprmidl 33184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-prmidl 33185 |
This theorem is referenced by: isprmidlc 33196 0ringprmidl 33198 rhmpreimaprmidl 33200 rsprprmprmidlb 33273 zarcls1 33511 zarclssn 33515 |
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