![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mxidlmaxv | Structured version Visualization version GIF version |
Description: An ideal πΌ strictly containing a maximal ideal π is the whole ring π΅. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
Ref | Expression |
---|---|
mxidlmaxv.1 | β’ π΅ = (Baseβπ ) |
mxidlmaxv.2 | β’ (π β π β Ring) |
mxidlmaxv.3 | β’ (π β π β (MaxIdealβπ )) |
mxidlmaxv.4 | β’ (π β πΌ β (LIdealβπ )) |
mxidlmaxv.5 | β’ (π β π β πΌ) |
mxidlmaxv.6 | β’ (π β π β (πΌ β π)) |
Ref | Expression |
---|---|
mxidlmaxv | β’ (π β πΌ = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mxidlmaxv.2 | . . 3 β’ (π β π β Ring) | |
2 | mxidlmaxv.3 | . . 3 β’ (π β π β (MaxIdealβπ )) | |
3 | mxidlmaxv.4 | . . 3 β’ (π β πΌ β (LIdealβπ )) | |
4 | mxidlmaxv.5 | . . 3 β’ (π β π β πΌ) | |
5 | mxidlmaxv.1 | . . . 4 β’ π΅ = (Baseβπ ) | |
6 | 5 | mxidlmax 33016 | . . 3 β’ (((π β Ring β§ π β (MaxIdealβπ )) β§ (πΌ β (LIdealβπ ) β§ π β πΌ)) β (πΌ = π β¨ πΌ = π΅)) |
7 | 1, 2, 3, 4, 6 | syl22anc 836 | . 2 β’ (π β (πΌ = π β¨ πΌ = π΅)) |
8 | mxidlmaxv.6 | . . . . 5 β’ (π β π β (πΌ β π)) | |
9 | 8 | eldifad 3952 | . . . 4 β’ (π β π β πΌ) |
10 | 8 | eldifbd 3953 | . . . 4 β’ (π β Β¬ π β π) |
11 | nelne1 3031 | . . . 4 β’ ((π β πΌ β§ Β¬ π β π) β πΌ β π) | |
12 | 9, 10, 11 | syl2anc 583 | . . 3 β’ (π β πΌ β π) |
13 | 12 | neneqd 2937 | . 2 β’ (π β Β¬ πΌ = π) |
14 | 7, 13 | orcnd 875 | 1 β’ (π β πΌ = π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β¨ wo 844 = wceq 1533 β wcel 2098 β wne 2932 β cdif 3937 β wss 3940 βcfv 6533 Basecbs 17142 Ringcrg 20127 LIdealclidl 21054 MaxIdealcmxidl 33010 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-mxidl 33011 |
This theorem is referenced by: mxidlirredi 33022 qsdrngilem 33043 |
Copyright terms: Public domain | W3C validator |