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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mxidlidl | Structured version Visualization version GIF version |
Description: A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.) |
Ref | Expression |
---|---|
mxidlval.1 | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
mxidlidl | β’ ((π β Ring β§ π β (MaxIdealβπ )) β π β (LIdealβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mxidlval.1 | . . . 4 β’ π΅ = (Baseβπ ) | |
2 | 1 | ismxidl 32286 | . . 3 β’ (π β Ring β (π β (MaxIdealβπ ) β (π β (LIdealβπ ) β§ π β π΅ β§ βπ β (LIdealβπ )(π β π β (π = π β¨ π = π΅))))) |
3 | 2 | biimpa 478 | . 2 β’ ((π β Ring β§ π β (MaxIdealβπ )) β (π β (LIdealβπ ) β§ π β π΅ β§ βπ β (LIdealβπ )(π β π β (π = π β¨ π = π΅)))) |
4 | 3 | simp1d 1143 | 1 β’ ((π β Ring β§ π β (MaxIdealβπ )) β π β (LIdealβπ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β¨ wo 846 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 β wss 3914 βcfv 6500 Basecbs 17091 Ringcrg 19972 LIdealclidl 20676 MaxIdealcmxidl 32283 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-mxidl 32284 |
This theorem is referenced by: mxidln1 32290 mxidlnzr 32291 mxidlprm 32292 zarclssn 32518 zarmxt1 32525 |
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