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Theorem mxidlirredi 33197
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 21257 . . . . 5 (𝜑𝑅 ∈ Ring)
4 mxidlirredi.1 . . . . 5 (𝜑𝑀 ∈ (MaxIdeal‘𝑅))
5 mxidlirredi.b . . . . . 6 𝐵 = (Base‘𝑅)
65mxidlnr 33190 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀𝐵)
73, 4, 6syl2anc 583 . . . 4 (𝜑𝑀𝐵)
8 eqid 2728 . . . . . 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 33152 . . . . 5 (𝜑 → (𝑀 = 𝐵𝑋 ∈ (Unit‘𝑅)))
1211necon3abid 2974 . . . 4 (𝜑 → (𝑀𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
137, 12mpbid 231 . . 3 (𝜑 → ¬ 𝑋 ∈ (Unit‘𝑅))
141, 13eldifd 3958 . 2 (𝜑𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)))
153ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑅 ∈ Ring)
164ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑀 ∈ (MaxIdeal‘𝑅))
17 simplr 768 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
1817eldifad 3959 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔𝐵)
1918snssd 4813 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → {𝑔} ⊆ 𝐵)
20 eqid 2728 . . . . . . . . . 10 (LIdeal‘𝑅) = (LIdeal‘𝑅)
219, 5, 20rspcl 21131 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
2215, 19, 21syl2anc 583 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) ∈ (LIdeal‘𝑅))
233ad4antr 731 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → 𝑅 ∈ Ring)
2423ad2antrr 725 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑅 ∈ Ring)
25 simp-5r 785 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))
2625eldifad 3959 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑔𝐵)
27 eqid 2728 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
28 simplr 768 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑞𝐵)
29 simp-6r 787 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
3029eldifad 3959 . . . . . . . . . . . . . 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 20199 . . . . . . . . . . . . 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 7427 . . . . . . . . . . . . . . 15 (𝑦 = (𝑞(.r𝑅)𝑓) → (𝑦(.r𝑅)𝑔) = ((𝑞(.r𝑅)𝑓)(.r𝑅)𝑔))
3332eqeq2d 2739 . . . . . . . . . . . . . 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 783 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → (𝑓(.r𝑅)𝑔) = 𝑋)
3635oveq2d 7436 . . . . . . . . . . . . . 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 20197 . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . 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 3611 . . . . . . . . . . . 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, 9rspsnel 33096 . . . . . . . . . . . . 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 837 . . . . . . . . . . 11 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑥 ∈ (𝐾‘{𝑔}))
441ad4antr 731 . . . . . . . . . . . 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 2839 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → 𝑥 ∈ (𝐾‘{𝑋}))
475, 27, 9rspsnel 33096 . . . . . . . . . . . . 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 837 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → ∃𝑞𝐵 𝑥 = (𝑞(.r𝑅)𝑋))
5043, 49r19.29a 3159 . . . . . . . . . 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 3986 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑀 ⊆ (𝐾‘{𝑔}))
539, 5rspssid 21132 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → {𝑔} ⊆ (𝐾‘{𝑔}))
54 vex 3475 . . . . . . . . . . . 12 𝑔 ∈ V
5554snss 4790 . . . . . . . . . . 11 (𝑔 ∈ (𝐾‘{𝑔}) ↔ {𝑔} ⊆ (𝐾‘{𝑔}))
5653, 55sylibr 233 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → 𝑔 ∈ (𝐾‘{𝑔}))
5715, 19, 56syl2anc 583 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ (𝐾‘{𝑔}))
58 df-idom 21232 . . . . . . . . . . . . . . 15 IDomn = (CRing ∩ Domn)
592, 58eleqtrdi 2839 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ (CRing ∩ Domn))
6059elin1d 4198 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ CRing)
6160ad6antr 735 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑅 ∈ CRing)
62 simplr 768 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑟𝐵)
63 simp-6r 787 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
6463eldifad 3959 . . . . . . . . . . . 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)
6618ad3antrrr 729 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑔𝐵)
67 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 )
6867oveq2d 7436 . . . . . . . . . . . . . . . . . 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 ))
69 simp-5r 785 . . . . . . . . . . . . . . . . . 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)
7015adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑅 ∈ Ring)
7170ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑅 ∈ Ring)
7264adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑓𝐵)
735, 27, 65ringrz 20230 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → (𝑓(.r𝑅) 0 ) = 0 )
7471, 72, 73syl2anc 583 . . . . . . . . . . . . . . . . . 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 )
7568, 69, 743eqtr3d 2776 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → 𝑋 = 0 )
76 mxidlirredi.y . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋0 )
7776neneqd 2942 . . . . . . . . . . . . . . . . . 18 (𝜑 → ¬ 𝑋 = 0 )
7877ad7antr 737 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) ∧ 𝑔 = 0 ) → ¬ 𝑋 = 0 )
7975, 78pm2.65da 816 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → ¬ 𝑔 = 0 )
8079neqned 2944 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑔0 )
81 eldifsn 4791 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝐵 ∖ { 0 }) ↔ (𝑔𝐵𝑔0 ))
8266, 80, 81sylanbrc 582 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑔 ∈ (𝐵 ∖ { 0 }))
8370ad2antrr 725 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑅 ∈ Ring)
845, 27, 83, 62, 64ringcld 20199 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑟(.r𝑅)𝑓) ∈ 𝐵)
85 eqid 2728 . . . . . . . . . . . . . . . . 17 (1r𝑅) = (1r𝑅)
865, 85ringidcl 20202 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝐵)
873, 86syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (1r𝑅) ∈ 𝐵)
8887ad6antr 735 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (1r𝑅) ∈ 𝐵)
892ad6antr 735 . . . . . . . . . . . . . 14 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑅 ∈ IDomn)
905, 27, 85, 83, 66ringlidmd 20208 . . . . . . . . . . . . . . 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, 83, 62, 64, 66ringassd 20197 . . . . . . . . . . . . . . . 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 783 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑓(.r𝑅)𝑔) = 𝑋)
9493oveq2d 7436 . . . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . 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, 82, 84, 88, 89, 96idomrcan 32962 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (𝑟(.r𝑅)𝑓) = (1r𝑅))
988, 851unit 20313 . . . . . . . . . . . . . . 15 (𝑅 ∈ Ring → (1r𝑅) ∈ (Unit‘𝑅))
993, 98syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1r𝑅) ∈ (Unit‘𝑅))
10099ad6antr 735 . . . . . . . . . . . . 13 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → (1r𝑅) ∈ (Unit‘𝑅))
10197, 100eqeltrd 2829 . . . . . . . . . . . 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 20320 . . . . . . . . . . . . 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 1371 . . . . . . . . . . 11 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑓 ∈ (Unit‘𝑅))
1051ad4antr 731 . . . . . . . . . . . 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 2839 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑔 ∈ (𝐾‘{𝑋}))
1085, 27, 9rspsnel 33096 . . . . . . . . . . . . 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))
11070, 105, 107, 109syl21anc 837 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → ∃𝑟𝐵 𝑔 = (𝑟(.r𝑅)𝑋))
111104, 110r19.29a 3159 . . . . . . . . . 10 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑓 ∈ (Unit‘𝑅))
112 simp-4r 783 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)))
113112eldifbd 3960 . . . . . . . . . 10 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → ¬ 𝑓 ∈ (Unit‘𝑅))
114111, 113pm2.65da 816 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → ¬ 𝑔𝑀)
11557, 114eldifd 3958 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ ((𝐾‘{𝑔}) ∖ 𝑀))
1165, 15, 16, 22, 52, 115mxidlmaxv 33194 . . . . . . 7 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) = 𝐵)
117 eqid 2728 . . . . . . . 8 (𝐾‘{𝑔}) = (𝐾‘{𝑔})
1182ad3antrrr 729 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑅 ∈ IDomn)
1198, 9, 117, 5, 18, 118unitpidl1 33152 . . . . . . 7 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → ((𝐾‘{𝑔}) = 𝐵𝑔 ∈ (Unit‘𝑅)))
120116, 119mpbid 231 . . . . . 6 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ (Unit‘𝑅))
12117eldifbd 3960 . . . . . 6 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → ¬ 𝑔 ∈ (Unit‘𝑅))
122120, 121pm2.65da 816 . . . . 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 2944 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → (𝑓(.r𝑅)𝑔) ≠ 𝑋)
125124ralrimivva 3197 . 2 (𝜑 → ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(.r𝑅)𝑔) ≠ 𝑋)
126 eqid 2728 . . 3 (Irred‘𝑅) = (Irred‘𝑅)
127 eqid 2728 . . 3 (𝐵 ∖ (Unit‘𝑅)) = (𝐵 ∖ (Unit‘𝑅))
1285, 8, 126, 127, 27isirred 20358 . 2 (𝑋 ∈ (Irred‘𝑅) ↔ (𝑋 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(.r𝑅)𝑔) ≠ 𝑋))
12914, 125, 128sylanbrc 582 1 (𝜑𝑋 ∈ (Irred‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2937  wral 3058  wrex 3067  cdif 3944  cin 3946  wss 3947  {csn 4629  cfv 6548  (class class class)co 7420  Basecbs 17180  .rcmulr 17234  0gc0g 17421  1rcur 20121  Ringcrg 20173  CRingccrg 20174  Unitcui 20294  Irredcir 20295  LIdealclidl 21102  RSpancrsp 21103  Domncdomn 21227  IDomncidom 21228  MaxIdealcmxidl 33185
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 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-ress 17210  df-plusg 17246  df-mulr 17247  df-sca 17249  df-vsca 17250  df-ip 17251  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893  df-minusg 18894  df-sbg 18895  df-subg 19078  df-cmn 19737  df-abl 19738  df-mgp 20075  df-rng 20093  df-ur 20122  df-ring 20175  df-cring 20176  df-oppr 20273  df-dvdsr 20296  df-unit 20297  df-irred 20298  df-invr 20327  df-nzr 20452  df-subrg 20508  df-lmod 20745  df-lss 20816  df-lsp 20856  df-sra 21058  df-rgmod 21059  df-lidl 21104  df-rsp 21105  df-domn 21231  df-idom 21232  df-mxidl 33186
This theorem is referenced by:  mxidlirred  33198
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