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Theorem cntzsdrg 36791
 Description: Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
cntzsdrg.b 𝐵 = (Base‘𝑅)
cntzsdrg.m 𝑀 = (mulGrp‘𝑅)
cntzsdrg.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzsdrg ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubDRing‘𝑅))

Proof of Theorem cntzsdrg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 472 . 2 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → 𝑅 ∈ DivRing)
2 drngring 18577 . . 3 (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
3 cntzsdrg.b . . . 4 𝐵 = (Base‘𝑅)
4 cntzsdrg.m . . . 4 𝑀 = (mulGrp‘𝑅)
5 cntzsdrg.z . . . 4 𝑍 = (Cntz‘𝑀)
63, 4, 5cntzsubr 18635 . . 3 ((𝑅 ∈ Ring ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubRing‘𝑅))
72, 6sylan 487 . 2 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubRing‘𝑅))
8 oveq2 6557 . . . . . . 7 (𝑦 = (0g𝑅) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)))
9 oveq1 6556 . . . . . . 7 (𝑦 = (0g𝑅) → (𝑦(.r𝑅)((invr𝑅)‘𝑥)) = ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥)))
108, 9eqeq12d 2625 . . . . . 6 (𝑦 = (0g𝑅) → ((((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)) ↔ (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)) = ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥))))
11 eldifsn 4260 . . . . . . . 8 (𝑦 ∈ (𝑆 ∖ {(0g𝑅)}) ↔ (𝑦𝑆𝑦 ≠ (0g𝑅)))
12 eqid 2610 . . . . . . . . . . . . . 14 (Unit‘𝑅) = (Unit‘𝑅)
134oveq1i 6559 . . . . . . . . . . . . . 14 (𝑀s (Unit‘𝑅)) = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))
14 eqid 2610 . . . . . . . . . . . . . 14 (invr𝑅) = (invr𝑅)
1512, 13, 14invrfval 18496 . . . . . . . . . . . . 13 (invr𝑅) = (invg‘(𝑀s (Unit‘𝑅)))
16 eqid 2610 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
173, 12, 16isdrng 18574 . . . . . . . . . . . . . . . 16 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ {(0g𝑅)})))
1817simprbi 479 . . . . . . . . . . . . . . 15 (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ {(0g𝑅)}))
1918oveq2d 6565 . . . . . . . . . . . . . 14 (𝑅 ∈ DivRing → (𝑀s (Unit‘𝑅)) = (𝑀s (𝐵 ∖ {(0g𝑅)})))
2019fveq2d 6107 . . . . . . . . . . . . 13 (𝑅 ∈ DivRing → (invg‘(𝑀s (Unit‘𝑅))) = (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
2115, 20syl5eq 2656 . . . . . . . . . . . 12 (𝑅 ∈ DivRing → (invr𝑅) = (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
2221ad2antrr 758 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → (invr𝑅) = (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
2322fveq1d 6105 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invr𝑅)‘𝑥) = ((invg‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘𝑥))
244oveq1i 6559 . . . . . . . . . . . . . 14 (𝑀s (𝐵 ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s (𝐵 ∖ {(0g𝑅)}))
253, 16, 24drngmgp 18582 . . . . . . . . . . . . 13 (𝑅 ∈ DivRing → (𝑀s (𝐵 ∖ {(0g𝑅)})) ∈ Grp)
2625ad2antrr 758 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → (𝑀s (𝐵 ∖ {(0g𝑅)})) ∈ Grp)
27 ssdif 3707 . . . . . . . . . . . . 13 (𝑆𝐵 → (𝑆 ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)}))
2827ad2antlr 759 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → (𝑆 ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)}))
29 difss 3699 . . . . . . . . . . . . . 14 (𝐵 ∖ {(0g𝑅)}) ⊆ 𝐵
30 eqid 2610 . . . . . . . . . . . . . . 15 (𝑀s (𝐵 ∖ {(0g𝑅)})) = (𝑀s (𝐵 ∖ {(0g𝑅)}))
314, 3mgpbas 18318 . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝑀)
3230, 31ressbas2 15758 . . . . . . . . . . . . . 14 ((𝐵 ∖ {(0g𝑅)}) ⊆ 𝐵 → (𝐵 ∖ {(0g𝑅)}) = (Base‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
3329, 32ax-mp 5 . . . . . . . . . . . . 13 (𝐵 ∖ {(0g𝑅)}) = (Base‘(𝑀s (𝐵 ∖ {(0g𝑅)})))
34 eqid 2610 . . . . . . . . . . . . 13 (Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)}))) = (Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))
3533, 34cntzsubg 17592 . . . . . . . . . . . 12 (((𝑀s (𝐵 ∖ {(0g𝑅)})) ∈ Grp ∧ (𝑆 ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)})) → ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) ∈ (SubGrp‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
3626, 28, 35syl2anc 691 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) ∈ (SubGrp‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
37 simpr 476 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → 𝑆𝐵)
38 difss 3699 . . . . . . . . . . . . . . . 16 (𝑆 ∖ {(0g𝑅)}) ⊆ 𝑆
3931, 5cntz2ss 17588 . . . . . . . . . . . . . . . 16 ((𝑆𝐵 ∧ (𝑆 ∖ {(0g𝑅)}) ⊆ 𝑆) → (𝑍𝑆) ⊆ (𝑍‘(𝑆 ∖ {(0g𝑅)})))
4037, 38, 39sylancl 693 . . . . . . . . . . . . . . 15 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ⊆ (𝑍‘(𝑆 ∖ {(0g𝑅)})))
4140ssdifssd 3710 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → ((𝑍𝑆) ∖ {(0g𝑅)}) ⊆ (𝑍‘(𝑆 ∖ {(0g𝑅)})))
4241sselda 3568 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ (𝑍‘(𝑆 ∖ {(0g𝑅)})))
4331, 5cntzssv 17584 . . . . . . . . . . . . . . 15 (𝑍𝑆) ⊆ 𝐵
44 ssdif 3707 . . . . . . . . . . . . . . 15 ((𝑍𝑆) ⊆ 𝐵 → ((𝑍𝑆) ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)}))
4543, 44mp1i 13 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → ((𝑍𝑆) ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)}))
4645sselda 3568 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ (𝐵 ∖ {(0g𝑅)}))
4742, 46elind 3760 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ ((𝑍‘(𝑆 ∖ {(0g𝑅)})) ∩ (𝐵 ∖ {(0g𝑅)})))
48 fvex 6113 . . . . . . . . . . . . . . 15 (Base‘𝑅) ∈ V
493, 48eqeltri 2684 . . . . . . . . . . . . . 14 𝐵 ∈ V
50 difexg 4735 . . . . . . . . . . . . . 14 (𝐵 ∈ V → (𝐵 ∖ {(0g𝑅)}) ∈ V)
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝐵 ∖ {(0g𝑅)}) ∈ V
5230, 5, 34resscntz 17587 . . . . . . . . . . . . 13 (((𝐵 ∖ {(0g𝑅)}) ∈ V ∧ (𝑆 ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)})) → ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) = ((𝑍‘(𝑆 ∖ {(0g𝑅)})) ∩ (𝐵 ∖ {(0g𝑅)})))
5351, 28, 52sylancr 694 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) = ((𝑍‘(𝑆 ∖ {(0g𝑅)})) ∩ (𝐵 ∖ {(0g𝑅)})))
5447, 53eleqtrrd 2691 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})))
55 eqid 2610 . . . . . . . . . . . 12 (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)}))) = (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)})))
5655subginvcl 17426 . . . . . . . . . . 11 ((((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) ∈ (SubGrp‘(𝑀s (𝐵 ∖ {(0g𝑅)}))) ∧ 𝑥 ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)}))) → ((invg‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘𝑥) ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})))
5736, 54, 56syl2anc 691 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invg‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘𝑥) ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})))
5823, 57eqeltrd 2688 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invr𝑅)‘𝑥) ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})))
59 eqid 2610 . . . . . . . . . . . . 13 (.r𝑅) = (.r𝑅)
604, 59mgpplusg 18316 . . . . . . . . . . . 12 (.r𝑅) = (+g𝑀)
6130, 60ressplusg 15818 . . . . . . . . . . 11 ((𝐵 ∖ {(0g𝑅)}) ∈ V → (.r𝑅) = (+g‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
6251, 61ax-mp 5 . . . . . . . . . 10 (.r𝑅) = (+g‘(𝑀s (𝐵 ∖ {(0g𝑅)})))
6362, 34cntzi 17585 . . . . . . . . 9 ((((invr𝑅)‘𝑥) ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) ∧ 𝑦 ∈ (𝑆 ∖ {(0g𝑅)})) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
6458, 63sylan 487 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦 ∈ (𝑆 ∖ {(0g𝑅)})) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
6511, 64sylan2br 492 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ (𝑦𝑆𝑦 ≠ (0g𝑅))) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
6665anassrs 678 . . . . . 6 (((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) ∧ 𝑦 ≠ (0g𝑅)) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
672ad3antrrr 762 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → 𝑅 ∈ Ring)
681adantr 480 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑅 ∈ DivRing)
69 eldifi 3694 . . . . . . . . . . . 12 (𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)}) → 𝑥 ∈ (𝑍𝑆))
7069adantl 481 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ (𝑍𝑆))
7143, 70sseldi 3566 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥𝐵)
72 eldifsni 4261 . . . . . . . . . . 11 (𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)}) → 𝑥 ≠ (0g𝑅))
7372adantl 481 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ≠ (0g𝑅))
743, 16, 14drnginvrcl 18587 . . . . . . . . . 10 ((𝑅 ∈ DivRing ∧ 𝑥𝐵𝑥 ≠ (0g𝑅)) → ((invr𝑅)‘𝑥) ∈ 𝐵)
7568, 71, 73, 74syl3anc 1318 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invr𝑅)‘𝑥) ∈ 𝐵)
7675adantr 480 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → ((invr𝑅)‘𝑥) ∈ 𝐵)
773, 59, 16ringrz 18411 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((invr𝑅)‘𝑥) ∈ 𝐵) → (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)) = (0g𝑅))
7867, 76, 77syl2anc 691 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)) = (0g𝑅))
793, 59, 16ringlz 18410 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((invr𝑅)‘𝑥) ∈ 𝐵) → ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥)) = (0g𝑅))
8067, 76, 79syl2anc 691 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥)) = (0g𝑅))
8178, 80eqtr4d 2647 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)) = ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥)))
8210, 66, 81pm2.61ne 2867 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
8382ralrimiva 2949 . . . 4 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ∀𝑦𝑆 (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
84 simplr 788 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑆𝐵)
8531, 60, 5cntzel 17579 . . . . 5 ((𝑆𝐵 ∧ ((invr𝑅)‘𝑥) ∈ 𝐵) → (((invr𝑅)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥))))
8684, 75, 85syl2anc 691 . . . 4 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → (((invr𝑅)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥))))
8783, 86mpbird 246 . . 3 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invr𝑅)‘𝑥) ∈ (𝑍𝑆))
8887ralrimiva 2949 . 2 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → ∀𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})((invr𝑅)‘𝑥) ∈ (𝑍𝑆))
8914, 16issdrg2 36787 . 2 ((𝑍𝑆) ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑍𝑆) ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})((invr𝑅)‘𝑥) ∈ (𝑍𝑆)))
901, 7, 88, 89syl3anbrc 1239 1 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubDRing‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246  SubGrpcsubg 17411  Cntzccntz 17571  mulGrpcmgp 18312  Ringcrg 18370  Unitcui 18462  invrcinvr 18494  DivRingcdr 18570  SubRingcsubrg 18599  SubDRingcsdrg 36784 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-cntz 17573  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-subrg 18601  df-sdrg 36785 This theorem is referenced by: (None)
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