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Theorem mxidlirred 33121
Description: In a principal ideal domain, maximal ideals are exactly the ideals generated by irreducible elements. (Contributed by Thierry Arnoux, 22-Mar-2025.)
Hypotheses
Ref Expression
mxidlirred.b 𝐵 = (Base‘𝑅)
mxidlirred.k 𝐾 = (RSpan‘𝑅)
mxidlirred.0 0 = (0g𝑅)
mxidlirred.m 𝑀 = (𝐾‘{𝑋})
mxidlirred.r (𝜑𝑅 ∈ PID)
mxidlirred.x (𝜑𝑋𝐵)
mxidlirred.y (𝜑𝑋0 )
mxidlirred.1 (𝜑𝑀 ∈ (LIdeal‘𝑅))
Assertion
Ref Expression
mxidlirred (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))

Proof of Theorem mxidlirred
Dummy variables 𝑡 𝑥 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mxidlirred.b . . 3 𝐵 = (Base‘𝑅)
2 mxidlirred.k . . 3 𝐾 = (RSpan‘𝑅)
3 mxidlirred.0 . . 3 0 = (0g𝑅)
4 mxidlirred.m . . 3 𝑀 = (𝐾‘{𝑋})
5 mxidlirred.r . . . . . 6 (𝜑𝑅 ∈ PID)
6 df-pid 21222 . . . . . 6 PID = (IDomn ∩ LPIR)
75, 6eleqtrdi 2838 . . . . 5 (𝜑𝑅 ∈ (IDomn ∩ LPIR))
87elin1d 4194 . . . 4 (𝜑𝑅 ∈ IDomn)
98adantr 480 . . 3 ((𝜑𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ IDomn)
10 mxidlirred.x . . . 4 (𝜑𝑋𝐵)
1110adantr 480 . . 3 ((𝜑𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋𝐵)
12 mxidlirred.y . . . 4 (𝜑𝑋0 )
1312adantr 480 . . 3 ((𝜑𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋0 )
14 simpr 484 . . 3 ((𝜑𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
151, 2, 3, 4, 9, 11, 13, 14mxidlirredi 33120 . 2 ((𝜑𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ (Irred‘𝑅))
16 eqid 2727 . . . . . . . . . . 11 (∥r𝑅) = (∥r𝑅)
17 simplr 768 . . . . . . . . . . . 12 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑥𝐵)
1817ad2antrr 725 . . . . . . . . . . 11 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑥𝐵)
1910ad8antr 739 . . . . . . . . . . 11 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑋𝐵)
20 eqid 2727 . . . . . . . . . . 11 (Unit‘𝑅) = (Unit‘𝑅)
21 eqid 2727 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
228idomringd 21244 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
2322ad4antr 731 . . . . . . . . . . . . 13 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑅 ∈ Ring)
2423ad2antrr 725 . . . . . . . . . . . 12 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑅 ∈ Ring)
2524ad2antrr 725 . . . . . . . . . . 11 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑅 ∈ Ring)
26 simplr 768 . . . . . . . . . . . . 13 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑡𝐵)
27 simpr 484 . . . . . . . . . . . . . 14 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑋 = (𝑡(.r𝑅)𝑥))
28 simp-8r 791 . . . . . . . . . . . . . 14 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑋 ∈ (Irred‘𝑅))
2927, 28eqeltrrd 2829 . . . . . . . . . . . . 13 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → (𝑡(.r𝑅)𝑥) ∈ (Irred‘𝑅))
30 eqid 2727 . . . . . . . . . . . . . 14 (Irred‘𝑅) = (Irred‘𝑅)
3130, 1, 20, 21irredmul 20357 . . . . . . . . . . . . 13 ((𝑡𝐵𝑥𝐵 ∧ (𝑡(.r𝑅)𝑥) ∈ (Irred‘𝑅)) → (𝑡 ∈ (Unit‘𝑅) ∨ 𝑥 ∈ (Unit‘𝑅)))
3226, 18, 29, 31syl3anc 1369 . . . . . . . . . . . 12 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → (𝑡 ∈ (Unit‘𝑅) ∨ 𝑥 ∈ (Unit‘𝑅)))
33 simpr 484 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑘 = (𝐾‘{𝑥}))
3433ad2antrr 725 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑘 = (𝐾‘{𝑥}))
35 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))
36 annim 403 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀𝑘 ∧ ¬ (𝑘 = 𝑀𝑘 = 𝐵)) ↔ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))
3735, 36sylibr 233 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → (𝑀𝑘 ∧ ¬ (𝑘 = 𝑀𝑘 = 𝐵)))
3837simprd 495 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → ¬ (𝑘 = 𝑀𝑘 = 𝐵))
39 ioran 982 . . . . . . . . . . . . . . . . . . 19 (¬ (𝑘 = 𝑀𝑘 = 𝐵) ↔ (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
4038, 39sylib 217 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
4140simprd 495 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → ¬ 𝑘 = 𝐵)
4241neqned 2942 . . . . . . . . . . . . . . . 16 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑘𝐵)
4342ad4antr 731 . . . . . . . . . . . . . . 15 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑘𝐵)
4434, 43eqnetrrd 3004 . . . . . . . . . . . . . 14 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → (𝐾‘{𝑥}) ≠ 𝐵)
4544neneqd 2940 . . . . . . . . . . . . 13 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → ¬ (𝐾‘{𝑥}) = 𝐵)
46 eqid 2727 . . . . . . . . . . . . . 14 (𝐾‘{𝑥}) = (𝐾‘{𝑥})
478ad8antr 739 . . . . . . . . . . . . . 14 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑅 ∈ IDomn)
4820, 2, 46, 1, 18, 47unitpidl1 33075 . . . . . . . . . . . . 13 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → ((𝐾‘{𝑥}) = 𝐵𝑥 ∈ (Unit‘𝑅)))
4945, 48mtbid 324 . . . . . . . . . . . 12 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → ¬ 𝑥 ∈ (Unit‘𝑅))
5032, 49olcnd 876 . . . . . . . . . . 11 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑡 ∈ (Unit‘𝑅))
5127eqcomd 2733 . . . . . . . . . . 11 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → (𝑡(.r𝑅)𝑥) = 𝑋)
521, 2, 16, 18, 19, 20, 21, 25, 50, 51dvdsruassoi 33028 . . . . . . . . . 10 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → (𝑥(∥r𝑅)𝑋𝑋(∥r𝑅)𝑥))
531, 2, 16, 18, 19, 25rspsnasso 33031 . . . . . . . . . 10 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → ((𝑥(∥r𝑅)𝑋𝑋(∥r𝑅)𝑥) ↔ (𝐾‘{𝑋}) = (𝐾‘{𝑥})))
5452, 53mpbid 231 . . . . . . . . 9 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → (𝐾‘{𝑋}) = (𝐾‘{𝑥}))
5554, 34eqtr4d 2770 . . . . . . . 8 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → (𝐾‘{𝑋}) = 𝑘)
564, 55eqtr2id 2780 . . . . . . 7 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑘 = 𝑀)
5740simpld 494 . . . . . . . 8 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → ¬ 𝑘 = 𝑀)
5857ad4antr 731 . . . . . . 7 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → ¬ 𝑘 = 𝑀)
5956, 58pm2.21dd 194 . . . . . 6 (((((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) ∧ 𝑡𝐵) ∧ 𝑋 = (𝑡(.r𝑅)𝑥)) → 𝑀 ∈ (MaxIdeal‘𝑅))
6037simpld 494 . . . . . . . . . 10 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑀𝑘)
6160ad2antrr 725 . . . . . . . . 9 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑀𝑘)
6210snssd 4808 . . . . . . . . . . . . 13 (𝜑 → {𝑋} ⊆ 𝐵)
632, 1rspssid 21121 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → {𝑋} ⊆ (𝐾‘{𝑋}))
6422, 62, 63syl2anc 583 . . . . . . . . . . . 12 (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
6564, 4sseqtrrdi 4029 . . . . . . . . . . 11 (𝜑 → {𝑋} ⊆ 𝑀)
66 snssg 4783 . . . . . . . . . . . 12 (𝑋𝐵 → (𝑋𝑀 ↔ {𝑋} ⊆ 𝑀))
6766biimpar 477 . . . . . . . . . . 11 ((𝑋𝐵 ∧ {𝑋} ⊆ 𝑀) → 𝑋𝑀)
6810, 65, 67syl2anc 583 . . . . . . . . . 10 (𝜑𝑋𝑀)
6968ad6antr 735 . . . . . . . . 9 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋𝑀)
7061, 69sseldd 3979 . . . . . . . 8 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋𝑘)
7170, 33eleqtrd 2830 . . . . . . 7 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋 ∈ (𝐾‘{𝑥}))
721, 21, 2rspsnel 33023 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑥𝐵) → (𝑋 ∈ (𝐾‘{𝑥}) ↔ ∃𝑡𝐵 𝑋 = (𝑡(.r𝑅)𝑥)))
7372biimpa 476 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑥𝐵) ∧ 𝑋 ∈ (𝐾‘{𝑥})) → ∃𝑡𝐵 𝑋 = (𝑡(.r𝑅)𝑥))
7424, 17, 71, 73syl21anc 837 . . . . . 6 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → ∃𝑡𝐵 𝑋 = (𝑡(.r𝑅)𝑥))
7559, 74r19.29a 3157 . . . . 5 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑀 ∈ (MaxIdeal‘𝑅))
76 simplr 768 . . . . . . 7 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑘 ∈ (LIdeal‘𝑅))
777elin2d 4195 . . . . . . . . 9 (𝜑𝑅 ∈ LPIR)
78 eqid 2727 . . . . . . . . . . 11 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
79 eqid 2727 . . . . . . . . . . 11 (LIdeal‘𝑅) = (LIdeal‘𝑅)
8078, 79islpir 21207 . . . . . . . . . 10 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
8180simprbi 496 . . . . . . . . 9 (𝑅 ∈ LPIR → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
8277, 81syl 17 . . . . . . . 8 (𝜑 → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
8382ad4antr 731 . . . . . . 7 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
8476, 83eleqtrd 2830 . . . . . 6 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑘 ∈ (LPIdeal‘𝑅))
8578, 2, 1islpidl 21204 . . . . . . 7 (𝑅 ∈ Ring → (𝑘 ∈ (LPIdeal‘𝑅) ↔ ∃𝑥𝐵 𝑘 = (𝐾‘{𝑥})))
8685biimpa 476 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑘 ∈ (LPIdeal‘𝑅)) → ∃𝑥𝐵 𝑘 = (𝐾‘{𝑥}))
8723, 84, 86syl2anc 583 . . . . 5 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → ∃𝑥𝐵 𝑘 = (𝐾‘{𝑥}))
8875, 87r19.29a 3157 . . . 4 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑀 ∈ (MaxIdeal‘𝑅))
89 mxidlirred.1 . . . . . . . 8 (𝜑𝑀 ∈ (LIdeal‘𝑅))
9089ad2antrr 725 . . . . . . 7 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
9130, 20irrednu 20353 . . . . . . . . . 10 (𝑋 ∈ (Irred‘𝑅) → ¬ 𝑋 ∈ (Unit‘𝑅))
9291adantl 481 . . . . . . . . 9 ((𝜑𝑋 ∈ (Irred‘𝑅)) → ¬ 𝑋 ∈ (Unit‘𝑅))
9320, 2, 4, 1, 10, 8unitpidl1 33075 . . . . . . . . . . 11 (𝜑 → (𝑀 = 𝐵𝑋 ∈ (Unit‘𝑅)))
9493adantr 480 . . . . . . . . . 10 ((𝜑𝑋 ∈ (Irred‘𝑅)) → (𝑀 = 𝐵𝑋 ∈ (Unit‘𝑅)))
9594necon3abid 2972 . . . . . . . . 9 ((𝜑𝑋 ∈ (Irred‘𝑅)) → (𝑀𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
9692, 95mpbird 257 . . . . . . . 8 ((𝜑𝑋 ∈ (Irred‘𝑅)) → 𝑀𝐵)
9796adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀𝐵)
9890, 97jca 511 . . . . . 6 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵))
991ismxidl 33111 . . . . . . . . . . 11 (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))))
10022, 99syl 17 . . . . . . . . . 10 (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))))
101 df-3an 1087 . . . . . . . . . 10 ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ↔ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))))
102100, 101bitrdi 287 . . . . . . . . 9 (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))))
103102notbid 318 . . . . . . . 8 (𝜑 → (¬ 𝑀 ∈ (MaxIdeal‘𝑅) ↔ ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))))
104103biimpa 476 . . . . . . 7 ((𝜑 ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))))
105104adantlr 714 . . . . . 6 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵) ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))))
10698, 105mpnanrd 409 . . . . 5 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))
107 rexnal 3095 . . . . 5 (∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)) ↔ ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))
108106, 107sylibr 233 . . . 4 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))
10988, 108r19.29a 3157 . . 3 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
110109pm2.18da 799 . 2 ((𝜑𝑋 ∈ (Irred‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
11115, 110impbida 800 1 (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  cin 3943  wss 3944  {csn 4624   class class class wbr 5142  cfv 6542  (class class class)co 7414  Basecbs 17171  .rcmulr 17225  0gc0g 17412  Ringcrg 20164  rcdsr 20282  Unitcui 20283  Irredcir 20284  LIdealclidl 21091  RSpancrsp 21092  LPIdealclpidl 21199  LPIRclpir 21200  IDomncidom 21217  PIDcpid 21218  MaxIdealcmxidl 33108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-sca 17240  df-vsca 17241  df-ip 17242  df-0g 17414  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-grp 18884  df-minusg 18885  df-sbg 18886  df-subg 19069  df-cmn 19728  df-abl 19729  df-mgp 20066  df-rng 20084  df-ur 20113  df-ring 20166  df-cring 20167  df-oppr 20262  df-dvdsr 20285  df-unit 20286  df-irred 20287  df-invr 20316  df-nzr 20441  df-subrg 20497  df-lmod 20734  df-lss 20805  df-lsp 20845  df-sra 21047  df-rgmod 21048  df-lidl 21093  df-rsp 21094  df-lpidl 21201  df-lpir 21202  df-domn 21220  df-idom 21221  df-pid 21222  df-mxidl 33109
This theorem is referenced by:  algextdeglem4  33324
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