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 20029 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
2 | eqid 2820 | . . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | eqid 2820 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 2, 3 | lidl0 19987 | . . . 4 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ NzRing → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
6 | fvex 6676 | . . . . . . 7 ⊢ (0g‘𝑅) ∈ V | |
7 | hashsng 13727 | . . . . . . 7 ⊢ ((0g‘𝑅) ∈ V → (♯‘{(0g‘𝑅)}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{(0g‘𝑅)}) = 1 |
9 | simpr 487 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → {(0g‘𝑅)} = (Base‘𝑅)) | |
10 | 9 | fveq2d 6667 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → (♯‘{(0g‘𝑅)}) = (♯‘(Base‘𝑅))) |
11 | 8, 10 | syl5eqr 2869 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 = (♯‘(Base‘𝑅))) |
12 | 1red 10635 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 ∈ ℝ) | |
13 | eqid 2820 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 13 | isnzr2hash 20032 | . . . . . . . . 9 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))) |
15 | 14 | simprbi 499 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 1 < (♯‘(Base‘𝑅))) |
16 | 15 | adantr 483 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 < (♯‘(Base‘𝑅))) |
17 | 12, 16 | ltned 10769 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → 1 ≠ (♯‘(Base‘𝑅))) |
18 | 17 | neneqd 3020 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ {(0g‘𝑅)} = (Base‘𝑅)) → ¬ 1 = (♯‘(Base‘𝑅))) |
19 | 11, 18 | pm2.65da 815 | . . . 4 ⊢ (𝑅 ∈ NzRing → ¬ {(0g‘𝑅)} = (Base‘𝑅)) |
20 | 19 | neqned 3022 | . . 3 ⊢ (𝑅 ∈ NzRing → {(0g‘𝑅)} ≠ (Base‘𝑅)) |
21 | 13 | ssmxidl 31003 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ {(0g‘𝑅)} ∈ (LIdeal‘𝑅) ∧ {(0g‘𝑅)} ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚) |
22 | 1, 5, 20, 21 | syl3anc 1366 | . 2 ⊢ (𝑅 ∈ NzRing → ∃𝑚 ∈ (MaxIdeal‘𝑅){(0g‘𝑅)} ⊆ 𝑚) |
23 | df-rex 3143 | . . 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 219 | . 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 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3015 ∃wrex 3138 Vcvv 3491 ⊆ wss 3929 {csn 4560 class class class wbr 5059 ‘cfv 6348 1c1 10531 < clt 10668 ♯chash 13687 Basecbs 16478 0gc0g 16708 Ringcrg 19292 LIdealclidl 19937 NzRingcnzr 20025 MaxIdealcmxidl 30992 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-ac2 9878 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-rpss 7442 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-dju 9323 df-card 9361 df-ac 9535 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12890 df-hash 13688 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-ip 16578 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-mgp 19235 df-ur 19247 df-ring 19294 df-subrg 19528 df-lmod 19631 df-lss 19699 df-sra 19939 df-rgmod 19940 df-lidl 19941 df-nzr 20026 df-mxidl 30993 |
This theorem is referenced by: mxidlnzrb 31005 |
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