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Theorem qsdrng 33525
Description: An ideal 𝑀 is both left and right maximal if and only if the factor ring 𝑄 is a division ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
Hypotheses
Ref Expression
qsdrng.0 𝑂 = (oppr𝑅)
qsdrng.q 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
qsdrng.r (𝜑𝑅 ∈ NzRing)
qsdrng.2 (𝜑𝑀 ∈ (2Ideal‘𝑅))
Assertion
Ref Expression
qsdrng (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))

Proof of Theorem qsdrng
Dummy variables 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsdrng.r . . . . . 6 (𝜑𝑅 ∈ NzRing)
2 nzrring 20516 . . . . . 6 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
31, 2syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
43adantr 480 . . . 4 ((𝜑𝑄 ∈ DivRing) → 𝑅 ∈ Ring)
5 qsdrng.2 . . . . . 6 (𝜑𝑀 ∈ (2Ideal‘𝑅))
652idllidld 21264 . . . . 5 (𝜑𝑀 ∈ (LIdeal‘𝑅))
76adantr 480 . . . 4 ((𝜑𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑅))
8 drngnzr 20748 . . . . . . 7 (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
98ad2antlr 727 . . . . . 6 (((𝜑𝑄 ∈ DivRing) ∧ 𝑀 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
10 qsdrng.q . . . . . . . . . . 11 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
11 eqid 2737 . . . . . . . . . . 11 (2Ideal‘𝑅) = (2Ideal‘𝑅)
1210, 11qusring 21285 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
133, 5, 12syl2anc 584 . . . . . . . . 9 (𝜑𝑄 ∈ Ring)
1413adantr 480 . . . . . . . 8 ((𝜑𝑀 = (Base‘𝑅)) → 𝑄 ∈ Ring)
15 oveq2 7439 . . . . . . . . . . . . . 14 (𝑀 = (Base‘𝑅) → (𝑅 ~QG 𝑀) = (𝑅 ~QG (Base‘𝑅)))
1615oveq2d 7447 . . . . . . . . . . . . 13 (𝑀 = (Base‘𝑅) → (𝑅 /s (𝑅 ~QG 𝑀)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
1710, 16eqtrid 2789 . . . . . . . . . . . 12 (𝑀 = (Base‘𝑅) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
1817fveq2d 6910 . . . . . . . . . . 11 (𝑀 = (Base‘𝑅) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
193ringgrpd 20239 . . . . . . . . . . . 12 (𝜑𝑅 ∈ Grp)
20 eqid 2737 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
21 eqid 2737 . . . . . . . . . . . . 13 (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
2220, 21qustriv 33392 . . . . . . . . . . . 12 (𝑅 ∈ Grp → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
2319, 22syl 17 . . . . . . . . . . 11 (𝜑 → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
2418, 23sylan9eqr 2799 . . . . . . . . . 10 ((𝜑𝑀 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
2524fveq2d 6910 . . . . . . . . 9 ((𝜑𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
26 fvex 6919 . . . . . . . . . 10 (Base‘𝑅) ∈ V
27 hashsng 14408 . . . . . . . . . 10 ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
2826, 27ax-mp 5 . . . . . . . . 9 (♯‘{(Base‘𝑅)}) = 1
2925, 28eqtrdi 2793 . . . . . . . 8 ((𝜑𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
30 0ringnnzr 20525 . . . . . . . . 9 (𝑄 ∈ Ring → ((♯‘(Base‘𝑄)) = 1 ↔ ¬ 𝑄 ∈ NzRing))
3130biimpa 476 . . . . . . . 8 ((𝑄 ∈ Ring ∧ (♯‘(Base‘𝑄)) = 1) → ¬ 𝑄 ∈ NzRing)
3214, 29, 31syl2anc 584 . . . . . . 7 ((𝜑𝑀 = (Base‘𝑅)) → ¬ 𝑄 ∈ NzRing)
3332adantlr 715 . . . . . 6 (((𝜑𝑄 ∈ DivRing) ∧ 𝑀 = (Base‘𝑅)) → ¬ 𝑄 ∈ NzRing)
349, 33pm2.65da 817 . . . . 5 ((𝜑𝑄 ∈ DivRing) → ¬ 𝑀 = (Base‘𝑅))
3534neqned 2947 . . . 4 ((𝜑𝑄 ∈ DivRing) → 𝑀 ≠ (Base‘𝑅))
36 simplr 769 . . . . . . . . . . 11 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀𝑗)
37 simpr 484 . . . . . . . . . . . . 13 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → ¬ 𝑗 = 𝑀)
3837neqned 2947 . . . . . . . . . . . 12 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗𝑀)
3938necomd 2996 . . . . . . . . . . 11 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀𝑗)
40 pssdifn0 4368 . . . . . . . . . . 11 ((𝑀𝑗𝑀𝑗) → (𝑗𝑀) ≠ ∅)
4136, 39, 40syl2anc 584 . . . . . . . . . 10 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → (𝑗𝑀) ≠ ∅)
42 n0 4353 . . . . . . . . . 10 ((𝑗𝑀) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑗𝑀))
4341, 42sylib 218 . . . . . . . . 9 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → ∃𝑥 𝑥 ∈ (𝑗𝑀))
44 qsdrng.0 . . . . . . . . . 10 𝑂 = (oppr𝑅)
451ad5antr 734 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑅 ∈ NzRing)
465ad5antr 734 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
47 simp-5r 786 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑄 ∈ DivRing)
48 simp-4r 784 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑗 ∈ (LIdeal‘𝑅))
4936adantr 480 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑀𝑗)
50 simpr 484 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑥 ∈ (𝑗𝑀))
5144, 10, 45, 46, 20, 47, 48, 49, 50qsdrnglem2 33524 . . . . . . . . 9 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑗 = (Base‘𝑅))
5243, 51exlimddv 1935 . . . . . . . 8 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 = (Base‘𝑅))
5352ex 412 . . . . . . 7 ((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) → (¬ 𝑗 = 𝑀𝑗 = (Base‘𝑅)))
5453orrd 864 . . . . . 6 ((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))
5554ex 412 . . . . 5 (((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
5655ralrimiva 3146 . . . 4 ((𝜑𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
5720ismxidl 33490 . . . . 5 (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))))
5857biimpar 477 . . . 4 ((𝑅 ∈ Ring ∧ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘𝑅))
594, 7, 35, 56, 58syl13anc 1374 . . 3 ((𝜑𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑅))
6044opprring 20347 . . . . . 6 (𝑅 ∈ Ring → 𝑂 ∈ Ring)
613, 60syl 17 . . . . 5 (𝜑𝑂 ∈ Ring)
6261adantr 480 . . . 4 ((𝜑𝑄 ∈ DivRing) → 𝑂 ∈ Ring)
635adantr 480 . . . . 5 ((𝜑𝑄 ∈ DivRing) → 𝑀 ∈ (2Ideal‘𝑅))
6463, 442idlridld 21265 . . . 4 ((𝜑𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑂))
65 simplr 769 . . . . . . . . . . 11 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀𝑗)
66 simpr 484 . . . . . . . . . . . . 13 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → ¬ 𝑗 = 𝑀)
6766neqned 2947 . . . . . . . . . . . 12 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗𝑀)
6867necomd 2996 . . . . . . . . . . 11 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀𝑗)
6965, 68, 40syl2anc 584 . . . . . . . . . 10 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → (𝑗𝑀) ≠ ∅)
7069, 42sylib 218 . . . . . . . . 9 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → ∃𝑥 𝑥 ∈ (𝑗𝑀))
71 eqid 2737 . . . . . . . . . 10 (oppr𝑂) = (oppr𝑂)
72 eqid 2737 . . . . . . . . . 10 (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
7344opprnzr 20522 . . . . . . . . . . . 12 (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
741, 73syl 17 . . . . . . . . . . 11 (𝜑𝑂 ∈ NzRing)
7574ad5antr 734 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑂 ∈ NzRing)
7644, 3oppr2idl 33514 . . . . . . . . . . . 12 (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
775, 76eleqtrd 2843 . . . . . . . . . . 11 (𝜑𝑀 ∈ (2Ideal‘𝑂))
7877ad5antr 734 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑀 ∈ (2Ideal‘𝑂))
7944, 20opprbas 20341 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑂)
80 eqid 2737 . . . . . . . . . . . . 13 (oppr𝑄) = (oppr𝑄)
8180opprdrng 20764 . . . . . . . . . . . 12 (𝑄 ∈ DivRing ↔ (oppr𝑄) ∈ DivRing)
8220, 44, 10, 3, 5opprqusdrng 33521 . . . . . . . . . . . . 13 (𝜑 → ((oppr𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing))
8382biimpa 476 . . . . . . . . . . . 12 ((𝜑 ∧ (oppr𝑄) ∈ DivRing) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
8481, 83sylan2b 594 . . . . . . . . . . 11 ((𝜑𝑄 ∈ DivRing) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
8584ad4antr 732 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
86 simp-4r 784 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑗 ∈ (LIdeal‘𝑂))
8765adantr 480 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑀𝑗)
88 simpr 484 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑥 ∈ (𝑗𝑀))
8971, 72, 75, 78, 79, 85, 86, 87, 88qsdrnglem2 33524 . . . . . . . . 9 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑗 = (Base‘𝑅))
9070, 89exlimddv 1935 . . . . . . . 8 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 = (Base‘𝑅))
9190ex 412 . . . . . . 7 ((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) → (¬ 𝑗 = 𝑀𝑗 = (Base‘𝑅)))
9291orrd 864 . . . . . 6 ((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))
9392ex 412 . . . . 5 (((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) → (𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
9493ralrimiva 3146 . . . 4 ((𝜑𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
9579ismxidl 33490 . . . . 5 (𝑂 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑂) ↔ (𝑀 ∈ (LIdeal‘𝑂) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))))
9695biimpar 477 . . . 4 ((𝑂 ∈ Ring ∧ (𝑀 ∈ (LIdeal‘𝑂) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘𝑂))
9762, 64, 35, 94, 96syl13anc 1374 . . 3 ((𝜑𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑂))
9859, 97jca 511 . 2 ((𝜑𝑄 ∈ DivRing) → (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂)))
991adantr 480 . . 3 ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑅 ∈ NzRing)
100 simprl 771 . . 3 ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑀 ∈ (MaxIdeal‘𝑅))
101 simprr 773 . . 3 ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑀 ∈ (MaxIdeal‘𝑂))
10244, 10, 99, 100, 101qsdrngi 33523 . 2 ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑄 ∈ DivRing)
10398, 102impbida 801 1 (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cdif 3948  wss 3951  c0 4333  {csn 4626  cfv 6561  (class class class)co 7431  1c1 11156  chash 14369  Basecbs 17247   /s cqus 17550  Grpcgrp 18951   ~QG cqg 19140  Ringcrg 20230  opprcoppr 20333  NzRingcnzr 20512  DivRingcdr 20729  LIdealclidl 21216  2Idealc2idl 21259  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-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  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-se 5638  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-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  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-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  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-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-imas 17553  df-qus 17554  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  df-cntz 19335  df-oppg 19364  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-nzr 20513  df-subrg 20570  df-drng 20731  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lmhm 21021  df-lbs 21074  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-dsmm 21752  df-frlm 21767  df-uvc 21803  df-mxidl 33488
This theorem is referenced by:  qsfld  33526
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