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Theorem qsdrng 33505
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 20533 . . . . . 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 21282 . . . . 5 (𝜑𝑀 ∈ (LIdeal‘𝑅))
76adantr 480 . . . 4 ((𝜑𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑅))
8 drngnzr 20765 . . . . . . 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 2735 . . . . . . . . . . 11 (2Ideal‘𝑅) = (2Ideal‘𝑅)
1210, 11qusring 21303 . . . . . . . . . 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 2787 . . . . . . . . . . . 12 (𝑀 = (Base‘𝑅) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
1817fveq2d 6911 . . . . . . . . . . 11 (𝑀 = (Base‘𝑅) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
193ringgrpd 20260 . . . . . . . . . . . 12 (𝜑𝑅 ∈ Grp)
20 eqid 2735 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
21 eqid 2735 . . . . . . . . . . . . 13 (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
2220, 21qustriv 33372 . . . . . . . . . . . 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 2797 . . . . . . . . . 10 ((𝜑𝑀 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
2524fveq2d 6911 . . . . . . . . 9 ((𝜑𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
26 fvex 6920 . . . . . . . . . 10 (Base‘𝑅) ∈ V
27 hashsng 14405 . . . . . . . . . 10 ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
2826, 27ax-mp 5 . . . . . . . . 9 (♯‘{(Base‘𝑅)}) = 1
2925, 28eqtrdi 2791 . . . . . . . 8 ((𝜑𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
30 0ringnnzr 20542 . . . . . . . . 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 2945 . . . 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 2945 . . . . . . . . . . . 12 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗𝑀)
3938necomd 2994 . . . . . . . . . . 11 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀𝑗)
40 pssdifn0 4374 . . . . . . . . . . 11 ((𝑀𝑗𝑀𝑗) → (𝑗𝑀) ≠ ∅)
4136, 39, 40syl2anc 584 . . . . . . . . . 10 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → (𝑗𝑀) ≠ ∅)
42 n0 4359 . . . . . . . . . 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 33504 . . . . . . . . 9 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑗 = (Base‘𝑅))
5243, 51exlimddv 1933 . . . . . . . 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 863 . . . . . 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 3144 . . . 4 ((𝜑𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
5720ismxidl 33470 . . . . 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 1371 . . 3 ((𝜑𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑅))
6044opprring 20364 . . . . . 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 21283 . . . 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 2945 . . . . . . . . . . . 12 (((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗𝑀)
6867necomd 2994 . . . . . . . . . . 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 2735 . . . . . . . . . 10 (oppr𝑂) = (oppr𝑂)
72 eqid 2735 . . . . . . . . . 10 (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
7344opprnzr 20539 . . . . . . . . . . . 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 33494 . . . . . . . . . . . 12 (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
775, 76eleqtrd 2841 . . . . . . . . . . 11 (𝜑𝑀 ∈ (2Ideal‘𝑂))
7877ad5antr 734 . . . . . . . . . 10 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑀 ∈ (2Ideal‘𝑂))
7944, 20opprbas 20358 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑂)
80 eqid 2735 . . . . . . . . . . . . 13 (oppr𝑄) = (oppr𝑄)
8180opprdrng 20781 . . . . . . . . . . . 12 (𝑄 ∈ DivRing ↔ (oppr𝑄) ∈ DivRing)
8220, 44, 10, 3, 5opprqusdrng 33501 . . . . . . . . . . . . 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 33504 . . . . . . . . 9 ((((((𝜑𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗𝑀)) → 𝑗 = (Base‘𝑅))
9070, 89exlimddv 1933 . . . . . . . 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 863 . . . . . 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 3144 . . . 4 ((𝜑𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
9579ismxidl 33470 . . . . 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 1371 . . 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 33503 . 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 847  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  Vcvv 3478  cdif 3960  wss 3963  c0 4339  {csn 4631  cfv 6563  (class class class)co 7431  1c1 11154  chash 14366  Basecbs 17245   /s cqus 17552  Grpcgrp 18964   ~QG cqg 19153  Ringcrg 20251  opprcoppr 20350  NzRingcnzr 20529  DivRingcdr 20746  LIdealclidl 21234  2Idealc2idl 21277  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-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  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-se 5642  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-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  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-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  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-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-imas 17555  df-qus 17556  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-cntz 19348  df-oppg 19377  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-nzr 20530  df-subrg 20587  df-drng 20748  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-2idl 21278  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-mxidl 33468
This theorem is referenced by:  qsfld  33506
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