Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cntzsdrg Structured version   Visualization version   GIF version

Theorem cntzsdrg 37288
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 473 . 2 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → 𝑅 ∈ DivRing)
2 drngring 18686 . . 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 18744 . . 3 ((𝑅 ∈ Ring ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubRing‘𝑅))
72, 6sylan 488 . 2 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubRing‘𝑅))
8 oveq2 6618 . . . . . . 7 (𝑦 = (0g𝑅) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)))
9 oveq1 6617 . . . . . . 7 (𝑦 = (0g𝑅) → (𝑦(.r𝑅)((invr𝑅)‘𝑥)) = ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥)))
108, 9eqeq12d 2636 . . . . . 6 (𝑦 = (0g𝑅) → ((((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)) ↔ (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)) = ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥))))
11 eldifsn 4292 . . . . . . . 8 (𝑦 ∈ (𝑆 ∖ {(0g𝑅)}) ↔ (𝑦𝑆𝑦 ≠ (0g𝑅)))
12 eqid 2621 . . . . . . . . . . . . . 14 (Unit‘𝑅) = (Unit‘𝑅)
134oveq1i 6620 . . . . . . . . . . . . . 14 (𝑀s (Unit‘𝑅)) = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))
14 eqid 2621 . . . . . . . . . . . . . 14 (invr𝑅) = (invr𝑅)
1512, 13, 14invrfval 18605 . . . . . . . . . . . . 13 (invr𝑅) = (invg‘(𝑀s (Unit‘𝑅)))
16 eqid 2621 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
173, 12, 16isdrng 18683 . . . . . . . . . . . . . . . 16 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ {(0g𝑅)})))
1817simprbi 480 . . . . . . . . . . . . . . 15 (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ {(0g𝑅)}))
1918oveq2d 6626 . . . . . . . . . . . . . 14 (𝑅 ∈ DivRing → (𝑀s (Unit‘𝑅)) = (𝑀s (𝐵 ∖ {(0g𝑅)})))
2019fveq2d 6157 . . . . . . . . . . . . 13 (𝑅 ∈ DivRing → (invg‘(𝑀s (Unit‘𝑅))) = (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
2115, 20syl5eq 2667 . . . . . . . . . . . 12 (𝑅 ∈ DivRing → (invr𝑅) = (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
2221ad2antrr 761 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → (invr𝑅) = (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
2322fveq1d 6155 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invr𝑅)‘𝑥) = ((invg‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘𝑥))
244oveq1i 6620 . . . . . . . . . . . . . 14 (𝑀s (𝐵 ∖ {(0g𝑅)})) = ((mulGrp‘𝑅) ↾s (𝐵 ∖ {(0g𝑅)}))
253, 16, 24drngmgp 18691 . . . . . . . . . . . . 13 (𝑅 ∈ DivRing → (𝑀s (𝐵 ∖ {(0g𝑅)})) ∈ Grp)
2625ad2antrr 761 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → (𝑀s (𝐵 ∖ {(0g𝑅)})) ∈ Grp)
27 ssdif 3728 . . . . . . . . . . . . 13 (𝑆𝐵 → (𝑆 ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)}))
2827ad2antlr 762 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → (𝑆 ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)}))
29 difss 3720 . . . . . . . . . . . . . 14 (𝐵 ∖ {(0g𝑅)}) ⊆ 𝐵
30 eqid 2621 . . . . . . . . . . . . . . 15 (𝑀s (𝐵 ∖ {(0g𝑅)})) = (𝑀s (𝐵 ∖ {(0g𝑅)}))
314, 3mgpbas 18427 . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝑀)
3230, 31ressbas2 15863 . . . . . . . . . . . . . 14 ((𝐵 ∖ {(0g𝑅)}) ⊆ 𝐵 → (𝐵 ∖ {(0g𝑅)}) = (Base‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
3329, 32ax-mp 5 . . . . . . . . . . . . 13 (𝐵 ∖ {(0g𝑅)}) = (Base‘(𝑀s (𝐵 ∖ {(0g𝑅)})))
34 eqid 2621 . . . . . . . . . . . . 13 (Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)}))) = (Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))
3533, 34cntzsubg 17701 . . . . . . . . . . . 12 (((𝑀s (𝐵 ∖ {(0g𝑅)})) ∈ Grp ∧ (𝑆 ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)})) → ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) ∈ (SubGrp‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
3626, 28, 35syl2anc 692 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) ∈ (SubGrp‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
37 simpr 477 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → 𝑆𝐵)
38 difss 3720 . . . . . . . . . . . . . . . 16 (𝑆 ∖ {(0g𝑅)}) ⊆ 𝑆
3931, 5cntz2ss 17697 . . . . . . . . . . . . . . . 16 ((𝑆𝐵 ∧ (𝑆 ∖ {(0g𝑅)}) ⊆ 𝑆) → (𝑍𝑆) ⊆ (𝑍‘(𝑆 ∖ {(0g𝑅)})))
4037, 38, 39sylancl 693 . . . . . . . . . . . . . . 15 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ⊆ (𝑍‘(𝑆 ∖ {(0g𝑅)})))
4140ssdifssd 3731 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → ((𝑍𝑆) ∖ {(0g𝑅)}) ⊆ (𝑍‘(𝑆 ∖ {(0g𝑅)})))
4241sselda 3587 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ (𝑍‘(𝑆 ∖ {(0g𝑅)})))
4331, 5cntzssv 17693 . . . . . . . . . . . . . . 15 (𝑍𝑆) ⊆ 𝐵
44 ssdif 3728 . . . . . . . . . . . . . . 15 ((𝑍𝑆) ⊆ 𝐵 → ((𝑍𝑆) ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)}))
4543, 44mp1i 13 . . . . . . . . . . . . . 14 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → ((𝑍𝑆) ∖ {(0g𝑅)}) ⊆ (𝐵 ∖ {(0g𝑅)}))
4645sselda 3587 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ (𝐵 ∖ {(0g𝑅)}))
4742, 46elind 3781 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ ((𝑍‘(𝑆 ∖ {(0g𝑅)})) ∩ (𝐵 ∖ {(0g𝑅)})))
48 fvex 6163 . . . . . . . . . . . . . . 15 (Base‘𝑅) ∈ V
493, 48eqeltri 2694 . . . . . . . . . . . . . 14 𝐵 ∈ V
50 difexg 4773 . . . . . . . . . . . . . 14 (𝐵 ∈ V → (𝐵 ∖ {(0g𝑅)}) ∈ V)
5149, 50ax-mp 5 . . . . . . . . . . . . 13 (𝐵 ∖ {(0g𝑅)}) ∈ V
5230, 5, 34resscntz 17696 . . . . . . . . . . . . 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 2701 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})))
55 eqid 2621 . . . . . . . . . . . 12 (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)}))) = (invg‘(𝑀s (𝐵 ∖ {(0g𝑅)})))
5655subginvcl 17535 . . . . . . . . . . 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 692 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invg‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘𝑥) ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})))
5823, 57eqeltrd 2698 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invr𝑅)‘𝑥) ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})))
59 eqid 2621 . . . . . . . . . . . . 13 (.r𝑅) = (.r𝑅)
604, 59mgpplusg 18425 . . . . . . . . . . . 12 (.r𝑅) = (+g𝑀)
6130, 60ressplusg 15925 . . . . . . . . . . 11 ((𝐵 ∖ {(0g𝑅)}) ∈ V → (.r𝑅) = (+g‘(𝑀s (𝐵 ∖ {(0g𝑅)}))))
6251, 61ax-mp 5 . . . . . . . . . 10 (.r𝑅) = (+g‘(𝑀s (𝐵 ∖ {(0g𝑅)})))
6362, 34cntzi 17694 . . . . . . . . 9 ((((invr𝑅)‘𝑥) ∈ ((Cntz‘(𝑀s (𝐵 ∖ {(0g𝑅)})))‘(𝑆 ∖ {(0g𝑅)})) ∧ 𝑦 ∈ (𝑆 ∖ {(0g𝑅)})) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
6458, 63sylan 488 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦 ∈ (𝑆 ∖ {(0g𝑅)})) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
6511, 64sylan2br 493 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ (𝑦𝑆𝑦 ≠ (0g𝑅))) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
6665anassrs 679 . . . . . 6 (((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) ∧ 𝑦 ≠ (0g𝑅)) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
672ad3antrrr 765 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → 𝑅 ∈ Ring)
681adantr 481 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑅 ∈ DivRing)
69 eldifi 3715 . . . . . . . . . . . 12 (𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)}) → 𝑥 ∈ (𝑍𝑆))
7069adantl 482 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ∈ (𝑍𝑆))
7143, 70sseldi 3585 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥𝐵)
72 eldifsni 4294 . . . . . . . . . . 11 (𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)}) → 𝑥 ≠ (0g𝑅))
7372adantl 482 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑥 ≠ (0g𝑅))
743, 16, 14drnginvrcl 18696 . . . . . . . . . 10 ((𝑅 ∈ DivRing ∧ 𝑥𝐵𝑥 ≠ (0g𝑅)) → ((invr𝑅)‘𝑥) ∈ 𝐵)
7568, 71, 73, 74syl3anc 1323 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invr𝑅)‘𝑥) ∈ 𝐵)
7675adantr 481 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → ((invr𝑅)‘𝑥) ∈ 𝐵)
773, 59, 16ringrz 18520 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((invr𝑅)‘𝑥) ∈ 𝐵) → (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)) = (0g𝑅))
7867, 76, 77syl2anc 692 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)) = (0g𝑅))
793, 59, 16ringlz 18519 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((invr𝑅)‘𝑥) ∈ 𝐵) → ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥)) = (0g𝑅))
8067, 76, 79syl2anc 692 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥)) = (0g𝑅))
8178, 80eqtr4d 2658 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → (((invr𝑅)‘𝑥)(.r𝑅)(0g𝑅)) = ((0g𝑅)(.r𝑅)((invr𝑅)‘𝑥)))
8210, 66, 81pm2.61ne 2875 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) ∧ 𝑦𝑆) → (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
8382ralrimiva 2961 . . . 4 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ∀𝑦𝑆 (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥)))
84 simplr 791 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → 𝑆𝐵)
8531, 60, 5cntzel 17688 . . . . 5 ((𝑆𝐵 ∧ ((invr𝑅)‘𝑥) ∈ 𝐵) → (((invr𝑅)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥))))
8684, 75, 85syl2anc 692 . . . 4 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → (((invr𝑅)‘𝑥) ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (((invr𝑅)‘𝑥)(.r𝑅)𝑦) = (𝑦(.r𝑅)((invr𝑅)‘𝑥))))
8783, 86mpbird 247 . . 3 (((𝑅 ∈ DivRing ∧ 𝑆𝐵) ∧ 𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})) → ((invr𝑅)‘𝑥) ∈ (𝑍𝑆))
8887ralrimiva 2961 . 2 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → ∀𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})((invr𝑅)‘𝑥) ∈ (𝑍𝑆))
8914, 16issdrg2 37284 . 2 ((𝑍𝑆) ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑍𝑆) ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ ((𝑍𝑆) ∖ {(0g𝑅)})((invr𝑅)‘𝑥) ∈ (𝑍𝑆)))
901, 7, 88, 89syl3anbrc 1244 1 ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubDRing‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3189  cdif 3556  cin 3558  wss 3559  {csn 4153  cfv 5852  (class class class)co 6610  Basecbs 15792  s cress 15793  +gcplusg 15873  .rcmulr 15874  0gc0g 16032  Grpcgrp 17354  invgcminusg 17355  SubGrpcsubg 17520  Cntzccntz 17680  mulGrpcmgp 18421  Ringcrg 18479  Unitcui 18571  invrcinvr 18603  DivRingcdr 18679  SubRingcsubrg 18708  SubDRingcsdrg 37281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-grp 17357  df-minusg 17358  df-subg 17523  df-cntz 17682  df-mgp 18422  df-ur 18434  df-ring 18481  df-oppr 18555  df-dvdsr 18573  df-unit 18574  df-invr 18604  df-dvr 18615  df-drng 18681  df-subrg 18710  df-sdrg 37282
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator