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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 33062 | . . 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 476 | . 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 395 β¨ wo 844 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 β wss 3943 βcfv 6537 (class class class)co 7405 Basecbs 17153 .rcmulr 17207 Ringcrg 20138 LIdealclidl 21065 PrmIdealcprmidl 33059 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-prmidl 33060 |
This theorem is referenced by: isprmidlc 33072 0ringprmidl 33074 rhmpreimaprmidl 33076 zarcls1 33379 zarclssn 33383 |
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