MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issubdrg Structured version   Visualization version   GIF version

Theorem issubdrg 19552
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 19530 . . . . . 6 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
41, 3syl 17 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑆 ∈ Ring)
5 simpr 487 . . . . . . . . 9 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (𝐴 ∖ { 0 }))
6 eldifsn 4711 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ { 0 }) ↔ (𝑥𝐴𝑥0 ))
75, 6sylib 220 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝑥𝐴𝑥0 ))
87simpld 497 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥𝐴)
92subrgbas 19536 . . . . . . . 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 2913 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ∈ (Base‘𝑆))
127simprd 498 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥0 )
13 issubdrg.z . . . . . . . . 9 0 = (0g𝑅)
142, 13subrg0 19534 . . . . . . . 8 (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g𝑆))
151, 14syl 17 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 0 = (0g𝑆))
1612, 15neeqtrd 3083 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → 𝑥 ≠ (0g𝑆))
17 eqid 2819 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
18 eqid 2819 . . . . . . . 8 (Unit‘𝑆) = (Unit‘𝑆)
19 eqid 2819 . . . . . . . 8 (0g𝑆) = (0g𝑆)
2017, 18, 19drngunit 19499 . . . . . . 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 2819 . . . . . 6 (invr𝑆) = (invr𝑆)
2418, 23, 17ringinvcl 19418 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝑆)) → ((invr𝑆)‘𝑥) ∈ (Base‘𝑆))
254, 22, 24syl2anc 586 . . . 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 19543 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ (Unit‘𝑆)) → (𝐼𝑥) = ((invr𝑆)‘𝑥))
281, 22, 27syl2anc 586 . . . 4 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼𝑥) = ((invr𝑆)‘𝑥))
2925, 28, 103eltr4d 2926 . . 3 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑆 ∈ DivRing) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → (𝐼𝑥) ∈ 𝐴)
3029ralrimiva 3180 . 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 2819 . . . . . . . . . 10 (Unit‘𝑅) = (Unit‘𝑅)
332, 32, 18subrguss 19542 . . . . . . . . 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 2819 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
3635, 32, 13isdrng 19498 . . . . . . . . . 10 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ { 0 })))
3736simprbi 499 . . . . . . . . 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 4005 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ ((Base‘𝑅) ∖ { 0 }))
4017, 18unitss 19402 . . . . . . . 8 (Unit‘𝑆) ⊆ (Base‘𝑆)
419ad2antlr 725 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → 𝐴 = (Base‘𝑆))
4240, 41sseqtrrid 4018 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ 𝐴)
4339, 42ssind 4207 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) ⊆ (((Base‘𝑅) ∖ { 0 }) ∩ 𝐴))
4435subrgss 19528 . . . . . . . 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 4264 . . . . . . 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 4006 . . . . 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 220 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → (𝑥𝐴𝑥0 ))
5251simpld 497 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥𝐴)
5349, 52sseldd 3966 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
5451simprd 498 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥0 )
5535, 32, 13drngunit 19499 . . . . . . . . . . . 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 19545 . . . . . . . . . . 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 1336 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ (𝐴 ∖ { 0 }) ∧ (𝐼𝑥) ∈ 𝐴)) → 𝑥 ∈ (Unit‘𝑆))
6261expr 459 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ (𝐴 ∖ { 0 })) → ((𝐼𝑥) ∈ 𝐴𝑥 ∈ (Unit‘𝑆)))
6362ralimdva 3175 . . . . . . 7 ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆)))
6463imp 409 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
65 dfss3 3954 . . . . . 6 ((𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆) ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })𝑥 ∈ (Unit‘𝑆))
6664, 65sylibr 236 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) ⊆ (Unit‘𝑆))
6748, 66eqssd 3982 . . . 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 4571 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → { 0 } = {(0g𝑆)})
7041, 69difeq12d 4098 . . . 4 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (𝐴 ∖ { 0 }) = ((Base‘𝑆) ∖ {(0g𝑆)}))
7167, 70eqtrd 2854 . . 3 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) ∧ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴) → (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g𝑆)}))
7217, 18, 19isdrng 19498 . . 3 (𝑆 ∈ DivRing ↔ (𝑆 ∈ Ring ∧ (Unit‘𝑆) = ((Base‘𝑆) ∖ {(0g𝑆)})))
7331, 71, 72sylanbrc 585 . 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 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wne 3014  wral 3136  cdif 3931  cin 3933  wss 3934  {csn 4559  cfv 6348  (class class class)co 7148  Basecbs 16475  s cress 16476  0gc0g 16705  Ringcrg 19289  Unitcui 19381  invrcinvr 19413  DivRingcdr 19494  SubRingcsubrg 19523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-tpos 7884  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-subg 18268  df-mgp 19232  df-ur 19244  df-ring 19291  df-oppr 19365  df-dvdsr 19383  df-unit 19384  df-invr 19414  df-drng 19496  df-subrg 19525
This theorem is referenced by:  issdrg2  19569  cnsubdrglem  20588  extdg1id  31046
  Copyright terms: Public domain W3C validator