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Theorem issubdrg 20247
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 774 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝐴 ∈ (SubRing‘𝑅))
2 issubdrg.s . . . . . . 7 𝑆 = (𝑅s 𝐴)
32subrgring 20225 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
41, 3syl 17 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑆 ∈ Ring)
5 simpr 485 . . . . . . . . 9 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (𝐴 ∖ { 0 }))
6 eldifsn 4747 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ { 0 }) ↔ (𝑥𝐴𝑥0 ))
75, 6sylib 217 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥𝐴𝑥0 ))
87simpld 495 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥𝐴)
92subrgbas 20231 . . . . . . . 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 2840 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Base‘𝑆))
127simprd 496 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥0 )
13 issubdrg.z . . . . . . . . 9 0 = (0g𝑅)
142, 13subrg0 20229 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g𝑆))
151, 14syl 17 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 0 = (0g𝑆))
1612, 15neeqtrd 3013 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ (0g𝑆))
17 eqid 2736 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
18 eqid 2736 . . . . . . . 8 (Unit‘𝑆) = (Unit‘𝑆)
19 eqid 2736 . . . . . . . 8 (0g𝑆) = (0g𝑆)
2017, 18, 19drngunit 20190 . . . . . . 7 (𝑆 ∈ DivRing → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))))
2120ad2antlr 725 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 ≠ (0g𝑆))))
2211, 16, 21mpbir2and 711 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Unit‘𝑆))
23 eqid 2736 . . . . . 6 (invr𝑆) = (invr𝑆)
2418, 23, 17ringinvcl 20105 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝑆)) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
254, 22, 24syl2anc 584 . . . 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 20238 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Unit‘𝑆)) → (𝐼𝑥) = ((invr𝑆)‘𝑥))
281, 22, 27syl2anc 584 . . . 4 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼𝑥) = ((invr𝑆)‘𝑥))
2925, 28, 103eltr4d 2853 . . 3 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼𝑥) ∈ 𝐴)
3029ralrimiva 3143 . 2 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) → ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴)
313ad2antlr 725 . . 3 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝑆 ∈ Ring)
32 eqid 2736 . . . . . . . . . 10 (Unit‘𝑅) = (Unit‘𝑅)
332, 32, 18subrguss 20237 . . . . . . . . 9 (𝐴 ∈ (SubRing‘𝑅) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
3433ad2antlr 725 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (Unit‘𝑅))
35 eqid 2736 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
3635, 32, 13isdrng 20189 . . . . . . . . . 10 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 })))
3736simprbi 497 . . . . . . . . 9 (𝑅 ∈ DivRing → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
3837ad2antrr 724 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 }))
3934, 38sseqtrd 3984 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ ((Base‘𝑅) ∖ { 0 }))
4017, 18unitss 20089 . . . . . . . 8 (Unit‘𝑆) ⊆ (Base‘𝑆)
419ad2antlr 725 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝐴 = (Base‘𝑆))
4240, 41sseqtrrid 3997 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ 𝐴)
4339, 42ssind 4192 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
4435subrgss 20223 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
4544ad2antlr 725 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝐴 ⊆ (Base‘𝑅))
46 difin2 4251 . . . . . . 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, 47sseqtrrd 3985 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (𝐴 ∖ { 0 }))
4944ad2antlr 725 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝐴 ⊆ (Base‘𝑅))
50 simprl 769 . . . . . . . . . . . . . 14 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (𝐴 ∖ { 0 }))
5150, 6sylib 217 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝑥𝐴𝑥0 ))
5251simpld 495 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥𝐴)
5349, 52sseldd 3945 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
5451simprd 496 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥0 )
5535, 32, 13drngunit 20190 . . . . . . . . . . . 12 (𝑅 ∈ DivRing → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥0 )))
5655ad2antrr 724 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝑥 ∈ (Unit‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑥0 )))
5753, 54, 56mpbir2and 711 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑅))
58 simprr 771 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝐼𝑥) ∈ 𝐴)
592, 32, 18, 26subrgunit 20240 . . . . . . . . . . 11 (𝐴 ∈ (SubRing‘𝑅) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥𝐴 ∧ (𝐼𝑥) ∈ 𝐴)))
6059ad2antlr 725 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥 ∈ (Unit‘𝑅) ∧ 𝑥𝐴 ∧ (𝐼𝑥) ∈ 𝐴)))
6157, 52, 58, 60mpbir3and 1342 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑆))
6261expr 457 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((𝐼𝑥) ∈ 𝐴𝑥 ∈ (Unit‘𝑆)))
6362ralimdva 3164 . . . . . . 7 ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆)))
6463imp 407 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
65 dfss3 3932 . . . . . 6 ((𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆) ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
6664, 65sylibr 233 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆))
6748, 66eqssd 3961 . . . 4 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) = (𝐴 ∖ { 0 }))
6814ad2antlr 725 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 0 = (0g𝑆))
6968sneqd 4598 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → { 0 } = {(0g𝑆)})
7041, 69difeq12d 4083 . . . 4 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = ((Base‘𝑆) ∖ {(0g𝑆)}))
7167, 70eqtrd 2776 . . 3 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g𝑆)}))
7217, 18, 19isdrng 20189 . . 3 (𝑆 ∈ DivRing ↔ (𝑆 ∈ Ring ∧ (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g𝑆)})))
7331, 71, 72sylanbrc 583 . 2 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝑆 ∈ DivRing)
7430, 73impbida 799 1 ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  cdif 3907  cin 3909  wss 3910  {csn 4586  cfv 6496  (class class class)co 7357  Basecbs 17083  s cress 17112  0gc0g 17321  Ringcrg 19964  Unitcui 20068  invrcinvr 20100  DivRingcdr 20185  SubRingcsubrg 20218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-drng 20187  df-subrg 20220
This theorem is referenced by:  issdrg2  20264  cnsubdrglem  20848  extdg1id  32352
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