Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > krull | Structured version Visualization version GIF version |
Description: Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
Ref | Expression |
---|---|
krull | ⊢ (𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nzrring 20027 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
2 | eqid 2798 | . . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | eqid 2798 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 2, 3 | lidl0 19985 | . . . 4 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ NzRing → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
6 | fvex 6658 | . . . . . . 7 ⊢ (0g‘𝑅) ∈ V | |
7 | hashsng 13726 | . . . . . . 7 ⊢ ((0g‘𝑅) ∈ V → (♯‘{(0g‘𝑅)}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{(0g‘𝑅)}) = 1 |
9 | simpr 488 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → {(0g‘𝑅)} = (Base‘𝑅)) | |
10 | 9 | fveq2d 6649 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → (♯‘{(0g‘𝑅)}) = (♯‘(Base‘𝑅))) |
11 | 8, 10 | syl5eqr 2847 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 = (♯‘(Base‘𝑅))) |
12 | 1red 10631 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 ∈ ℝ) | |
13 | eqid 2798 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 13 | isnzr2hash 20030 | . . . . . . . . 9 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))) |
15 | 14 | simprbi 500 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 1 < (♯‘(Base‘𝑅))) |
16 | 15 | adantr 484 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 < (♯‘(Base‘𝑅))) |
17 | 12, 16 | ltned 10765 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 ≠ (♯‘(Base‘𝑅))) |
18 | 17 | neneqd 2992 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → ¬ 1 = (♯‘(Base‘𝑅))) |
19 | 11, 18 | pm2.65da 816 | . . . 4 ⊢ (𝑅 ∈ NzRing → ¬ {(0g‘𝑅)} = (Base‘𝑅)) |
20 | 19 | neqned 2994 | . . 3 ⊢ (𝑅 ∈ NzRing → {(0g‘𝑅)} ≠ (Base‘𝑅)) |
21 | 13 | ssmxidl 31050 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ {(0g‘𝑅)} ∈ (LIdeal‘𝑅) ∧ {(0g‘𝑅)} ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚) |
22 | 1, 5, 20, 21 | syl3anc 1368 | . 2 ⊢ (𝑅 ∈ NzRing → ∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚) |
23 | df-rex 3112 | . . 3 ⊢ (∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚 ↔ ∃𝑚(𝑚 ∈ (MaxIdeal‘𝑅) ∧ {(0g‘𝑅)} ⊆ 𝑚)) | |
24 | exsimpl 1869 | . . 3 ⊢ (∃𝑚(𝑚 ∈ (MaxIdeal‘𝑅) ∧ {(0g‘𝑅)} ⊆ 𝑚) → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)) | |
25 | 23, 24 | sylbi 220 | . 2 ⊢ (∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚 → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)) |
26 | 22, 25 | syl 17 | 1 ⊢ (𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 {csn 4525 class class class wbr 5030 ‘cfv 6324 1c1 10527 < clt 10664 ♯chash 13686 Basecbs 16475 0gc0g 16705 Ringcrg 19290 LIdealclidl 19935 NzRingcnzr 20023 MaxIdealcmxidl 31039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-rpss 7429 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-nzr 20024 df-mxidl 31040 |
This theorem is referenced by: mxidlnzrb 31052 |
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