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Theorem issubdrg 18628
Description: Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
Hypotheses
Ref Expression
issubdrg.s 𝑆 = (𝑅s 𝐴)
issubdrg.z 0 = (0g𝑅)
issubdrg.i 𝐼 = (invr𝑅)
Assertion
Ref Expression
issubdrg ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅   𝑥,𝑆   𝑥, 0
Allowed substitution hint:   𝐼(𝑥)

Proof of Theorem issubdrg
StepHypRef Expression
1 simpllr 795 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝐴 ∈ (SubRing‘𝑅))
2 issubdrg.s . . . . . . 7 𝑆 = (𝑅s 𝐴)
32subrgring 18606 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
41, 3syl 17 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑆 ∈ Ring)
5 simpr 476 . . . . . . . . 9 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (𝐴 ∖ { 0 }))
6 eldifsn 4260 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ { 0 }) ↔ (𝑥𝐴𝑥0 ))
75, 6sylib 207 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥𝐴𝑥0 ))
87simpld 474 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥𝐴)
92subrgbas 18612 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
101, 9syl 17 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝐴 = (Base‘𝑆))
118, 10eleqtrd 2690 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Base‘𝑆))
127simprd 478 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥0 )
13 issubdrg.z . . . . . . . . 9 0 = (0g𝑅)
142, 13subrg0 18610 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g𝑆))
151, 14syl 17 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 0 = (0g𝑆))
1612, 15neeqtrd 2851 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ (0g𝑆))
17 eqid 2610 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
18 eqid 2610 . . . . . . . 8 (Unit‘𝑆) = (Unit‘𝑆)
19 eqid 2610 . . . . . . . 8 (0g𝑆) = (0g𝑆)
2017, 18, 19drngunit 18575 . . . . . . 7 (𝑆 ∈ DivRing → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))))
2120ad2antlr 759 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))))
2211, 16, 21mpbir2and 959 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Unit‘𝑆))
23 eqid 2610 . . . . . 6 (invr𝑆) = (invr𝑆)
2418, 23, 17ringinvcl 18499 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝑆)) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
254, 22, 24syl2anc 691 . . . 4 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
26 issubdrg.i . . . . . 6 𝐼 = (invr𝑅)
272, 26, 18, 23subrginv 18619 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Unit‘𝑆)) → (𝐼𝑥) = ((invr𝑆)‘𝑥))
281, 22, 27syl2anc 691 . . . 4 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼𝑥) = ((invr𝑆)‘𝑥))
2925, 28, 103eltr4d 2703 . . 3 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼𝑥) ∈ 𝐴)
3029ralrimiva 2949 . 2 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) → ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴)
313ad2antlr 759 . . 3 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝑆 ∈ Ring)
32 eqid 2610 . . . . . . . . . 10 (Unit‘𝑅) = (Unit‘𝑅)
332, 32, 18subrguss 18618 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
3433ad2antlr 759 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
35 eqid 2610 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
3635, 32, 13isdrng 18574 . . . . . . . . . 10 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 })))
3736simprbi 479 . . . . . . . . 9 (𝑅 ∈ DivRing → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
3837ad2antrr 758 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
3934, 38sseqtrd 3604 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ ((Base‘𝑅) ∖ { 0 }))
4017, 18unitss 18483 . . . . . . . 8 (Unit‘𝑆) ⊆ (Base‘𝑆)
419ad2antlr 759 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝐴 = (Base‘𝑆))
4240, 41syl5sseqr 3617 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ 𝐴)
4339, 42ssind 3799 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
4435subrgss 18604 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
4544ad2antlr 759 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝐴 ⊆ (Base‘𝑅))
46 difin2 3849 . . . . . . 7 (𝐴 ⊆ (Base‘𝑅) → (𝐴 ∖ { 0 }) = (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
4745, 46syl 17 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
4843, 47sseqtr4d 3605 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (𝐴 ∖ { 0 }))
4944ad2antlr 759 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝐴 ⊆ (Base‘𝑅))
50 simprl 790 . . . . . . . . . . . . . 14 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (𝐴 ∖ { 0 }))
5150, 6sylib 207 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝑥𝐴𝑥0 ))
5251simpld 474 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥𝐴)
5349, 52sseldd 3569 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
5451simprd 478 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥0 )
5535, 32, 13drngunit 18575 . . . . . . . . . . . 12 (𝑅 ∈ DivRing → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥0 )))
5655ad2antrr 758 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥0 )))
5753, 54, 56mpbir2and 959 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑅))
58 simprr 792 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝐼𝑥) ∈ 𝐴)
592, 32, 18, 26subrgunit 18621 . . . . . . . . . . 11 (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥𝐴 ∧ (𝐼𝑥) ∈ 𝐴)))
6059ad2antlr 759 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥𝐴 ∧ (𝐼𝑥) ∈ 𝐴)))
6157, 52, 58, 60mpbir3and 1238 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑆))
6261expr 641 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((𝐼𝑥) ∈ 𝐴𝑥 ∈ (Unit‘𝑆)))
6362ralimdva 2945 . . . . . . 7 ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆)))
6463imp 444 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
65 dfss3 3558 . . . . . 6 ((𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆) ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
6664, 65sylibr 223 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆))
6748, 66eqssd 3585 . . . 4 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) = (𝐴 ∖ { 0 }))
6814ad2antlr 759 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 0 = (0g𝑆))
6968sneqd 4137 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → { 0 } = {(0g𝑆)})
7041, 69difeq12d 3691 . . . 4 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = ((Base‘𝑆) ∖ {(0g𝑆)}))
7167, 70eqtrd 2644 . . 3 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g𝑆)}))
7217, 18, 19isdrng 18574 . . 3 (𝑆 ∈ DivRing ↔ (𝑆 ∈ Ring ∧ (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g𝑆)})))
7331, 71, 72sylanbrc 695 . 2 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝑆 ∈ DivRing)
7430, 73impbida 873 1 ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  cdif 3537  cin 3539  wss 3540  {csn 4125  cfv 5804  (class class class)co 6549  Basecbs 15695  s cress 15696  0gc0g 15923  Ringcrg 18370  Unitcui 18462  invrcinvr 18494  DivRingcdr 18570  SubRingcsubrg 18599
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-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-grp 17248  df-minusg 17249  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-drng 18572  df-subrg 18601
This theorem is referenced by:  cnsubdrglem  19616  issdrg2  36787
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