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Theorem mxidlirredi 33499
Description: In an integral domain, the generator of a maximal ideal is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.)
Hypotheses
Ref Expression
mxidlirredi.b 𝐵 = (Base‘𝑅)
mxidlirredi.k 𝐾 = (RSpan‘𝑅)
mxidlirredi.0 0 = (0g𝑅)
mxidlirredi.m 𝑀 = (𝐾‘{𝑋})
mxidlirredi.r (𝜑𝑅 ∈ IDomn)
mxidlirredi.x (𝜑𝑋𝐵)
mxidlirredi.y (𝜑𝑋0 )
mxidlirredi.1 (𝜑𝑀 ∈ (MaxIdeal‘𝑅))
Assertion
Ref Expression
mxidlirredi (𝜑𝑋 ∈ (Irred‘𝑅))

Proof of Theorem mxidlirredi
Dummy variables 𝑓 𝑔 𝑞 𝑥 𝑦 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mxidlirredi.x . . 3 (𝜑𝑋𝐵)
2 mxidlirredi.r . . . . . 6 (𝜑𝑅 ∈ IDomn)
32idomringd 20728 . . . . 5 (𝜑𝑅 ∈ Ring)
4 mxidlirredi.1 . . . . 5 (𝜑𝑀 ∈ (MaxIdeal‘𝑅))
5 mxidlirredi.b . . . . . 6 𝐵 = (Base‘𝑅)
65mxidlnr 33492 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀𝐵)
73, 4, 6syl2anc 584 . . . 4 (𝜑𝑀𝐵)
8 eqid 2737 . . . . . 6 (Unit‘𝑅) = (Unit‘𝑅)
9 mxidlirredi.k . . . . . 6 𝐾 = (RSpan‘𝑅)
10 mxidlirredi.m . . . . . 6 𝑀 = (𝐾‘{𝑋})
118, 9, 10, 5, 1, 2unitpidl1 33452 . . . . 5 (𝜑 → (𝑀 = 𝐵𝑋 ∈ (Unit‘𝑅)))
1211necon3abid 2977 . . . 4 (𝜑 → (𝑀𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
137, 12mpbid 232 . . 3 (𝜑 → ¬ 𝑋 ∈ (Unit‘𝑅))
141, 13eldifd 3962 . 2 (𝜑𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)))
153ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑅 ∈ Ring)
164ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑀 ∈ (MaxIdeal‘𝑅))
17 simplr 769 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
1817eldifad 3963 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔𝐵)
1918snssd 4809 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → {𝑔} ⊆ 𝐵)
20 eqid 2737 . . . . . . . . . 10 (LIdeal‘𝑅) = (LIdeal‘𝑅)
219, 5, 20rspcl 21245 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
2215, 19, 21syl2anc 584 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
233ad4antr 732 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → 𝑅 ∈ Ring)
2423ad2antrr 726 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑅 ∈ Ring)
25 simp-5r 786 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
2625eldifad 3963 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑔𝐵)
27 eqid 2737 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
28 simplr 769 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑞𝐵)
29 simp-6r 788 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
3029eldifad 3963 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑓𝐵)
315, 27, 24, 28, 30ringcld 20257 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → (𝑞(.r𝑅)𝑓) ∈ 𝐵)
32 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑦 = (𝑞(.r𝑅)𝑓) → (𝑦(.r𝑅)𝑔) = ((𝑞(.r𝑅)𝑓)(.r𝑅)𝑔))
3332eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑦 = (𝑞(.r𝑅)𝑓) → (𝑥 = (𝑦(.r𝑅)𝑔) ↔ 𝑥 = ((𝑞(.r𝑅)𝑓)(.r𝑅)𝑔)))
3433adantl 481 . . . . . . . . . . . . 13 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) ∧ 𝑦 = (𝑞(.r𝑅)𝑓)) → (𝑥 = (𝑦(.r𝑅)𝑔) ↔ 𝑥 = ((𝑞(.r𝑅)𝑓)(.r𝑅)𝑔)))
35 simp-4r 784 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → (𝑓(.r𝑅)𝑔) = 𝑋)
3635oveq2d 7447 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → (𝑞(.r𝑅)(𝑓(.r𝑅)𝑔)) = (𝑞(.r𝑅)𝑋))
375, 27, 24, 28, 30, 26ringassd 20254 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → ((𝑞(.r𝑅)𝑓)(.r𝑅)𝑔) = (𝑞(.r𝑅)(𝑓(.r𝑅)𝑔)))
38 simpr 484 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑥 = (𝑞(.r𝑅)𝑋))
3936, 37, 383eqtr4rd 2788 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑥 = ((𝑞(.r𝑅)𝑓)(.r𝑅)𝑔))
4031, 34, 39rspcedvd 3624 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → ∃𝑦𝐵 𝑥 = (𝑦(.r𝑅)𝑔))
415, 27, 9elrspsn 21250 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑔𝐵) → (𝑥 ∈ (𝐾‘{𝑔}) ↔ ∃𝑦𝐵 𝑥 = (𝑦(.r𝑅)𝑔)))
4241biimpar 477 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑔𝐵) ∧ ∃𝑦𝐵 𝑥 = (𝑦(.r𝑅)𝑔)) → 𝑥 ∈ (𝐾‘{𝑔}))
4324, 26, 40, 42syl21anc 838 . . . . . . . . . . 11 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑥 ∈ (𝐾‘{𝑔}))
441ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → 𝑋𝐵)
45 simpr 484 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → 𝑥𝑀)
4645, 10eleqtrdi 2851 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → 𝑥 ∈ (𝐾‘{𝑋}))
475, 27, 9elrspsn 21250 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑥 ∈ (𝐾‘{𝑋}) ↔ ∃𝑞𝐵 𝑥 = (𝑞(.r𝑅)𝑋)))
4847biimpa 476 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑋𝐵) ∧ 𝑥 ∈ (𝐾‘{𝑋})) → ∃𝑞𝐵 𝑥 = (𝑞(.r𝑅)𝑋))
4923, 44, 46, 48syl21anc 838 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → ∃𝑞𝐵 𝑥 = (𝑞(.r𝑅)𝑋))
5043, 49r19.29a 3162 . . . . . . . . . 10 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → 𝑥 ∈ (𝐾‘{𝑔}))
5150ex 412 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → (𝑥𝑀𝑥 ∈ (𝐾‘{𝑔})))
5251ssrdv 3989 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑀 ⊆ (𝐾‘{𝑔}))
539, 5rspssid 21246 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → {𝑔} ⊆ (𝐾‘{𝑔}))
54 vex 3484 . . . . . . . . . . . 12 𝑔 ∈ V
5554snss 4785 . . . . . . . . . . 11 (𝑔 ∈ (𝐾‘{𝑔}) ↔ {𝑔} ⊆ (𝐾‘{𝑔}))
5653, 55sylibr 234 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → 𝑔 ∈ (𝐾‘{𝑔}))
5715, 19, 56syl2anc 584 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐾‘{𝑔}))
58 df-idom 20696 . . . . . . . . . . . . . . 15 IDomn = (CRing ∩ Domn)
592, 58eleqtrdi 2851 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ (CRing ∩ Domn))
6059elin1d 4204 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ CRing)
6160ad6antr 736 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑅 ∈ CRing)
62 simplr 769 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑟𝐵)
63 simp-6r 788 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
6463eldifad 3963 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑓𝐵)
65 mxidlirredi.0 . . . . . . . . . . . . . 14 0 = (0g𝑅)
6615adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑅 ∈ Ring)
6766ad2antrr 726 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑅 ∈ Ring)
685, 27, 67, 62, 64ringcld 20257 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑟(.r𝑅)𝑓) ∈ 𝐵)
69 eqid 2737 . . . . . . . . . . . . . . . . 17 (1r𝑅) = (1r𝑅)
705, 69ringidcl 20262 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝐵)
713, 70syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (1r𝑅) ∈ 𝐵)
7271ad6antr 736 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (1r𝑅) ∈ 𝐵)
7318ad3antrrr 730 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑔𝐵)
74 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑔 = 0 )
7574oveq2d 7447 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(.r𝑅)𝑔) = (𝑓(.r𝑅) 0 ))
76 simp-5r 786 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(.r𝑅)𝑔) = 𝑋)
7766ad3antrrr 730 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑅 ∈ Ring)
7864adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑓𝐵)
795, 27, 65ringrz 20291 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → (𝑓(.r𝑅) 0 ) = 0 )
8077, 78, 79syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → (𝑓(.r𝑅) 0 ) = 0 )
8175, 76, 803eqtr3d 2785 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑋 = 0 )
82 mxidlirredi.y . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋0 )
8382neneqd 2945 . . . . . . . . . . . . . . . . . 18 (𝜑 → ¬ 𝑋 = 0 )
8483ad7antr 738 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → ¬ 𝑋 = 0 )
8581, 84pm2.65da 817 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → ¬ 𝑔 = 0 )
8685neqned 2947 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑔0 )
87 eldifsn 4786 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐵 ∖ { 0 }) ↔ (𝑔𝐵𝑔0 ))
8873, 86, 87sylanbrc 583 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑔 ∈ (𝐵 ∖ { 0 }))
892ad6antr 736 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑅 ∈ IDomn)
905, 27, 69, 67, 73ringlidmd 20269 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → ((1r𝑅)(.r𝑅)𝑔) = 𝑔)
91 simpr 484 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑔 = (𝑟(.r𝑅)𝑋))
925, 27, 67, 62, 64, 73ringassd 20254 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → ((𝑟(.r𝑅)𝑓)(.r𝑅)𝑔) = (𝑟(.r𝑅)(𝑓(.r𝑅)𝑔)))
93 simp-4r 784 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑓(.r𝑅)𝑔) = 𝑋)
9493oveq2d 7447 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑟(.r𝑅)(𝑓(.r𝑅)𝑔)) = (𝑟(.r𝑅)𝑋))
9592, 94eqtr2d 2778 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑟(.r𝑅)𝑋) = ((𝑟(.r𝑅)𝑓)(.r𝑅)𝑔))
9690, 91, 953eqtrrd 2782 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → ((𝑟(.r𝑅)𝑓)(.r𝑅)𝑔) = ((1r𝑅)(.r𝑅)𝑔))
975, 65, 27, 68, 72, 88, 89, 96idomrcan 33282 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑟(.r𝑅)𝑓) = (1r𝑅))
988, 691unit 20374 . . . . . . . . . . . . . . 15 (𝑅 ∈ Ring → (1r𝑅) ∈ (Unit‘𝑅))
993, 98syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1r𝑅) ∈ (Unit‘𝑅))
10099ad6antr 736 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (1r𝑅) ∈ (Unit‘𝑅))
10197, 100eqeltrd 2841 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑟(.r𝑅)𝑓) ∈ (Unit‘𝑅))
1028, 27, 5unitmulclb 20381 . . . . . . . . . . . . 13 ((𝑅 ∈ CRing ∧ 𝑟𝐵𝑓𝐵) → ((𝑟(.r𝑅)𝑓) ∈ (Unit‘𝑅) ↔ (𝑟 ∈ (Unit‘𝑅) ∧ 𝑓 ∈ (Unit‘𝑅))))
103102simplbda 499 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑟𝐵𝑓𝐵) ∧ (𝑟(.r𝑅)𝑓) ∈ (Unit‘𝑅)) → 𝑓 ∈ (Unit‘𝑅))
10461, 62, 64, 101, 103syl31anc 1375 . . . . . . . . . . 11 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑓 ∈ (Unit‘𝑅))
1051ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑋𝐵)
106 simpr 484 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑔𝑀)
107106, 10eleqtrdi 2851 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑔 ∈ (𝐾‘{𝑋}))
1085, 27, 9elrspsn 21250 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑔 ∈ (𝐾‘{𝑋}) ↔ ∃𝑟𝐵 𝑔 = (𝑟(.r𝑅)𝑋)))
109108biimpa 476 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝑋𝐵) ∧ 𝑔 ∈ (𝐾‘{𝑋})) → ∃𝑟𝐵 𝑔 = (𝑟(.r𝑅)𝑋))
11066, 105, 107, 109syl21anc 838 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → ∃𝑟𝐵 𝑔 = (𝑟(.r𝑅)𝑋))
111104, 110r19.29a 3162 . . . . . . . . . 10 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑓 ∈ (Unit‘𝑅))
112 simp-4r 784 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
113112eldifbd 3964 . . . . . . . . . 10 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → ¬ 𝑓 ∈ (Unit‘𝑅))
114111, 113pm2.65da 817 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → ¬ 𝑔𝑀)
11557, 114eldifd 3962 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ ((𝐾‘{𝑔}) ∖ 𝑀))
1165, 15, 16, 22, 52, 115mxidlmaxv 33496 . . . . . . 7 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) = 𝐵)
117 eqid 2737 . . . . . . . 8 (𝐾‘{𝑔}) = (𝐾‘{𝑔})
1182ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑅 ∈ IDomn)
1198, 9, 117, 5, 18, 118unitpidl1 33452 . . . . . . 7 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → ((𝐾‘{𝑔}) = 𝐵𝑔 ∈ (Unit‘𝑅)))
120116, 119mpbid 232 . . . . . 6 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ (Unit‘𝑅))
12117eldifbd 3964 . . . . . 6 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ (Unit‘𝑅))
122120, 121pm2.65da 817 . . . . 5 (((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) → ¬ (𝑓(.r𝑅)𝑔) = 𝑋)
123122anasss 466 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → ¬ (𝑓(.r𝑅)𝑔) = 𝑋)
124123neqned 2947 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → (𝑓(.r𝑅)𝑔) ≠ 𝑋)
125124ralrimivva 3202 . 2 (𝜑 → ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(.r𝑅)𝑔) ≠ 𝑋)
126 eqid 2737 . . 3 (Irred‘𝑅) = (Irred‘𝑅)
127 eqid 2737 . . 3 (𝐵 ∖ (Unit‘𝑅)) = (𝐵 ∖ (Unit‘𝑅))
1285, 8, 126, 127, 27isirred 20419 . 2 (𝑋 ∈ (Irred‘𝑅) ↔ (𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(.r𝑅)𝑔) ≠ 𝑋))
12914, 125, 128sylanbrc 583 1 (𝜑𝑋 ∈ (Irred‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cdif 3948  cin 3950  wss 3951  {csn 4626  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  0gc0g 17484  1rcur 20178  Ringcrg 20230  CRingccrg 20231  Unitcui 20355  Irredcir 20356  Domncdomn 20692  IDomncidom 20693  LIdealclidl 21216  RSpancrsp 21217  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-mxidl 33488
This theorem is referenced by:  mxidlirred  33500
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