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Theorem mxidlirredi 33479
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 20745 . . . . 5 (𝜑𝑅 ∈ Ring)
4 mxidlirredi.1 . . . . 5 (𝜑𝑀 ∈ (MaxIdeal‘𝑅))
5 mxidlirredi.b . . . . . 6 𝐵 = (Base‘𝑅)
65mxidlnr 33472 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀𝐵)
73, 4, 6syl2anc 584 . . . 4 (𝜑𝑀𝐵)
8 eqid 2735 . . . . . 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 33432 . . . . 5 (𝜑 → (𝑀 = 𝐵𝑋 ∈ (Unit‘𝑅)))
1211necon3abid 2975 . . . 4 (𝜑 → (𝑀𝐵 ↔ ¬ 𝑋 ∈ (Unit‘𝑅)))
137, 12mpbid 232 . . 3 (𝜑 → ¬ 𝑋 ∈ (Unit‘𝑅))
141, 13eldifd 3974 . 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 3975 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔𝐵)
1918snssd 4814 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → {𝑔} ⊆ 𝐵)
20 eqid 2735 . . . . . . . . . 10 (LIdeal‘𝑅) = (LIdeal‘𝑅)
219, 5, 20rspcl 21263 . . . . . . . . 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 3975 . . . . . . . . . . . 12 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) ∧ 𝑞𝐵) ∧ 𝑥 = (𝑞(.r𝑅)𝑋)) → 𝑔𝐵)
27 eqid 2735 . . . . . . . . . . . . . 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 3975 . . . . . . . . . . . . . 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 20277 . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . 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 20275 . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . 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 21268 . . . . . . . . . . . . 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 2849 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑥𝑀) → 𝑥 ∈ (𝐾‘{𝑋}))
475, 27, 9elrspsn 21268 . . . . . . . . . . . . 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 3160 . . . . . . . . . 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 4001 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑀 ⊆ (𝐾‘{𝑔}))
539, 5rspssid 21264 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ {𝑔} ⊆ 𝐵) → {𝑔} ⊆ (𝐾‘{𝑔}))
54 vex 3482 . . . . . . . . . . . 12 𝑔 ∈ V
5554snss 4790 . . . . . . . . . . 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 20713 . . . . . . . . . . . . . . 15 IDomn = (CRing ∩ Domn)
592, 58eleqtrdi 2849 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ (CRing ∩ Domn))
6059elin1d 4214 . . . . . . . . . . . . 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 3975 . . . . . . . . . . . 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 20277 . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (1r𝑅) = (1r𝑅)
705, 69ringidcl 20280 . . . . . . . . . . . . . . . 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 20308 . . . . . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . . . . 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 2943 . . . . . . . . . . . . . . . . . 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 2945 . . . . . . . . . . . . . . 15 (((((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) ∧ 𝑟𝐵) ∧ 𝑔 = (𝑟(.r𝑅)𝑋)) → 𝑔0 )
87 eldifsn 4791 . . . . . . . . . . . . . . 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 20286 . . . . . . . . . . . . . . 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 20275 . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . 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 33263 . . . . . . . . . . . . 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 20391 . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . 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 20398 . . . . . . . . . . . . 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 1372 . . . . . . . . . . 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 2849 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) ∧ 𝑔𝑀) → 𝑔 ∈ (𝐾‘{𝑋}))
1085, 27, 9elrspsn 21268 . . . . . . . . . . . . 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 3160 . . . . . . . . . 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 3976 . . . . . . . . . 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 3974 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → 𝑔 ∈ ((𝐾‘{𝑔}) ∖ 𝑀))
1165, 15, 16, 22, 52, 115mxidlmaxv 33476 . . . . . . 7 ((((𝜑𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))) ∧ (𝑓(.r𝑅)𝑔) = 𝑋) → (𝐾‘{𝑔}) = 𝐵)
117 eqid 2735 . . . . . . . 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 33432 . . . . . . 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 3976 . . . . . 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 2945 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝐵 ∖ (Unit‘𝑅)) ∧ 𝑔 ∈ (𝐵 ∖ (Unit‘𝑅)))) → (𝑓(.r𝑅)𝑔) ≠ 𝑋)
125124ralrimivva 3200 . 2 (𝜑 → ∀𝑓 ∈ (𝐵 ∖ (Unit‘𝑅))∀𝑔 ∈ (𝐵 ∖ (Unit‘𝑅))(𝑓(.r𝑅)𝑔) ≠ 𝑋)
126 eqid 2735 . . 3 (Irred‘𝑅) = (Irred‘𝑅)
127 eqid 2735 . . 3 (𝐵 ∖ (Unit‘𝑅)) = (𝐵 ∖ (Unit‘𝑅))
1285, 8, 126, 127, 27isirred 20436 . 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 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cdif 3960  cin 3962  wss 3963  {csn 4631  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  0gc0g 17486  1rcur 20199  Ringcrg 20251  CRingccrg 20252  Unitcui 20372  Irredcir 20373  Domncdomn 20709  IDomncidom 20710  LIdealclidl 21234  RSpancrsp 21235  MaxIdealcmxidl 33467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-irred 20376  df-invr 20405  df-nzr 20530  df-subrg 20587  df-domn 20712  df-idom 20713  df-lmod 20877  df-lss 20948  df-lsp 20988  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-mxidl 33468
This theorem is referenced by:  mxidlirred  33480
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