| 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 20516 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
| 2 | eqid 2737 | . . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 4 | 2, 3 | lidl0 21240 | . . . 4 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
| 5 | 1, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ NzRing → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
| 6 | fvex 6919 | . . . . . . 7 ⊢ (0g‘𝑅) ∈ V | |
| 7 | hashsng 14408 | . . . . . . 7 ⊢ ((0g‘𝑅) ∈ V → (♯‘{(0g‘𝑅)}) = 1) | |
| 8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{(0g‘𝑅)}) = 1 |
| 9 | simpr 484 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → {(0g‘𝑅)} = (Base‘𝑅)) | |
| 10 | 9 | fveq2d 6910 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → (♯‘{(0g‘𝑅)}) = (♯‘(Base‘𝑅))) |
| 11 | 8, 10 | eqtr3id 2791 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 = (♯‘(Base‘𝑅))) |
| 12 | 1red 11262 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 ∈ ℝ) | |
| 13 | eqid 2737 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 14 | 13 | isnzr2hash 20519 | . . . . . . . . 9 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))) |
| 15 | 14 | simprbi 496 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 1 < (♯‘(Base‘𝑅))) |
| 16 | 15 | adantr 480 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 < (♯‘(Base‘𝑅))) |
| 17 | 12, 16 | ltned 11397 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 ≠ (♯‘(Base‘𝑅))) |
| 18 | 17 | neneqd 2945 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → ¬ 1 = (♯‘(Base‘𝑅))) |
| 19 | 11, 18 | pm2.65da 817 | . . . 4 ⊢ (𝑅 ∈ NzRing → ¬ {(0g‘𝑅)} = (Base‘𝑅)) |
| 20 | 19 | neqned 2947 | . . 3 ⊢ (𝑅 ∈ NzRing → {(0g‘𝑅)} ≠ (Base‘𝑅)) |
| 21 | 13 | ssmxidl 33502 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ {(0g‘𝑅)} ∈ (LIdeal‘𝑅) ∧ {(0g‘𝑅)} ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚) |
| 22 | 1, 5, 20, 21 | syl3anc 1373 | . 2 ⊢ (𝑅 ∈ NzRing → ∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚) |
| 23 | df-rex 3071 | . . 3 ⊢ (∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚 ↔ ∃𝑚(𝑚 ∈ (MaxIdeal‘𝑅) ∧ {(0g‘𝑅)} ⊆ 𝑚)) | |
| 24 | exsimpl 1868 | . . 3 ⊢ (∃𝑚(𝑚 ∈ (MaxIdeal‘𝑅) ∧ {(0g‘𝑅)} ⊆ 𝑚) → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)) | |
| 25 | 23, 24 | sylbi 217 | . 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 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 {csn 4626 class class class wbr 5143 ‘cfv 6561 1c1 11156 < clt 11295 ♯chash 14369 Basecbs 17247 0gc0g 17484 Ringcrg 20230 NzRingcnzr 20512 LIdealclidl 21216 MaxIdealcmxidl 33487 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rpss 7743 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-nzr 20513 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-mxidl 33488 |
| This theorem is referenced by: mxidlnzrb 33508 krullndrng 33509 |
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