Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mxidlirred Structured version   Visualization version   GIF version

Theorem mxidlirred 33500
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 21347 . . . . . 6 PID = (IDomn ∩ LPIR)
75, 6eleqtrdi 2851 . . . . 5 (𝜑𝑅 ∈ (IDomn ∩ LPIR))
87elin1d 4204 . . . 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 33499 . 2 ((𝜑𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑋 ∈ (Irred‘𝑅))
16 eqid 2737 . . . . . . . . . . 11 (∥r𝑅) = (∥r𝑅)
17 simplr 769 . . . . . . . . . . . 12 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑥𝐵)
1817ad2antrr 726 . . . . . . . . . . 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 740 . . . . . . . . . . 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 2737 . . . . . . . . . . 11 (Unit‘𝑅) = (Unit‘𝑅)
21 eqid 2737 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
228idomringd 20728 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
2322ad4antr 732 . . . . . . . . . . . . 13 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑅 ∈ Ring)
2423ad2antrr 726 . . . . . . . . . . . 12 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑅 ∈ Ring)
2524ad2antrr 726 . . . . . . . . . . 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 769 . . . . . . . . . . . . 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 792 . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (Irred‘𝑅) = (Irred‘𝑅)
3130, 1, 20, 21irredmul 20429 . . . . . . . . . . . . 13 ((𝑡𝐵𝑥𝐵 ∧ (𝑡(.r𝑅)𝑥) ∈ (Irred‘𝑅)) → (𝑡 ∈ (Unit‘𝑅) ∨ 𝑥 ∈ (Unit‘𝑅)))
3226, 18, 29, 31syl3anc 1373 . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . . . . 19 (¬ (𝑘 = 𝑀𝑘 = 𝐵) ↔ (¬ 𝑘 = 𝑀 ∧ ¬ 𝑘 = 𝐵))
4038, 39sylib 218 . . . . . . . . . . . . . . . . . 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 2947 . . . . . . . . . . . . . . . 16 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑘𝐵)
4342ad4antr 732 . . . . . . . . . . . . . . 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 3009 . . . . . . . . . . . . . 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 2945 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (𝐾‘{𝑥}) = (𝐾‘{𝑥})
478ad8antr 740 . . . . . . . . . . . . . 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 33452 . . . . . . . . . . . . 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 878 . . . . . . . . . . 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 2743 . . . . . . . . . . 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 33412 . . . . . . . . . 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 33416 . . . . . . . . . 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 232 . . . . . . . . 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 2780 . . . . . . . 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 2790 . . . . . . 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 732 . . . . . . 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 195 . . . . . 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 726 . . . . . . . . 9 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑀𝑘)
6210snssd 4809 . . . . . . . . . . . . 13 (𝜑 → {𝑋} ⊆ 𝐵)
632, 1rspssid 21246 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → {𝑋} ⊆ (𝐾‘{𝑋}))
6422, 62, 63syl2anc 584 . . . . . . . . . . . 12 (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋}))
6564, 4sseqtrrdi 4025 . . . . . . . . . . 11 (𝜑 → {𝑋} ⊆ 𝑀)
66 snssg 4783 . . . . . . . . . . . 12 (𝑋𝐵 → (𝑋𝑀 ↔ {𝑋} ⊆ 𝑀))
6766biimpar 477 . . . . . . . . . . 11 ((𝑋𝐵 ∧ {𝑋} ⊆ 𝑀) → 𝑋𝑀)
6810, 65, 67syl2anc 584 . . . . . . . . . 10 (𝜑𝑋𝑀)
6968ad6antr 736 . . . . . . . . 9 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋𝑀)
7061, 69sseldd 3984 . . . . . . . 8 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋𝑘)
7170, 33eleqtrd 2843 . . . . . . 7 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑋 ∈ (𝐾‘{𝑥}))
721, 21, 2elrspsn 21250 . . . . . . . 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 838 . . . . . 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 3162 . . . . 5 (((((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) ∧ 𝑥𝐵) ∧ 𝑘 = (𝐾‘{𝑥})) → 𝑀 ∈ (MaxIdeal‘𝑅))
76 simplr 769 . . . . . . 7 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑘 ∈ (LIdeal‘𝑅))
777elin2d 4205 . . . . . . . . 9 (𝜑𝑅 ∈ LPIR)
78 eqid 2737 . . . . . . . . . . 11 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
79 eqid 2737 . . . . . . . . . . 11 (LIdeal‘𝑅) = (LIdeal‘𝑅)
8078, 79islpir 21338 . . . . . . . . . 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 732 . . . . . . 7 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → (LIdeal‘𝑅) = (LPIdeal‘𝑅))
8476, 83eleqtrd 2843 . . . . . 6 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → 𝑘 ∈ (LPIdeal‘𝑅))
8578, 2, 1islpidl 21335 . . . . . . 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 584 . . . . 5 (((((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵))) → ∃𝑥𝐵 𝑘 = (𝐾‘{𝑥}))
8875, 87r19.29a 3162 . . . 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 726 . . . . . . 7 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
9130, 20irrednu 20425 . . . . . . . . . 10 (𝑋 ∈ (Irred‘𝑅) → ¬ 𝑋 ∈ (Unit‘𝑅))
9291adantl 481 . . . . . . . . 9 ((𝜑𝑋 ∈ (Irred‘𝑅)) → ¬ 𝑋 ∈ (Unit‘𝑅))
9320, 2, 4, 1, 10, 8unitpidl1 33452 . . . . . . . . . . 11 (𝜑 → (𝑀 = 𝐵𝑋 ∈ (Unit‘𝑅)))
9493adantr 480 . . . . . . . . . 10 ((𝜑𝑋 ∈ (Irred‘𝑅)) → (𝑀 = 𝐵𝑋 ∈ (Unit‘𝑅)))
9594necon3abid 2977 . . . . . . . . 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 33490 . . . . . . . . . . 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 1089 . . . . . . . . . 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 715 . . . . . 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 3100 . . . . 5 (∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)) ↔ ¬ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))
108106, 107sylibr 234 . . . 4 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∃𝑘 ∈ (LIdeal‘𝑅) ¬ (𝑀𝑘 → (𝑘 = 𝑀𝑘 = 𝐵)))
10988, 108r19.29a 3162 . . 3 (((𝜑𝑋 ∈ (Irred‘𝑅)) ∧ ¬ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
110109pm2.18da 800 . 2 ((𝜑𝑋 ∈ (Irred‘𝑅)) → 𝑀 ∈ (MaxIdeal‘𝑅))
11115, 110impbida 801 1 (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cin 3950  wss 3951  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  0gc0g 17484  Ringcrg 20230  rcdsr 20354  Unitcui 20355  Irredcir 20356  IDomncidom 20693  LIdealclidl 21216  RSpancrsp 21217  LPIdealclpidl 21330  LPIRclpir 21331  PIDcpid 21346  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-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-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-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  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-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-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-irred 20359  df-invr 20388  df-nzr 20513  df-subrg 20570  df-domn 20695  df-idom 20696  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-lpidl 21332  df-lpir 21333  df-pid 21347  df-mxidl 33488
This theorem is referenced by:  rprmirredb  33560  algextdeglem4  33761
  Copyright terms: Public domain W3C validator