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Theorem issubdrg 20631
Description: Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
Hypotheses
Ref Expression
issubdrg.s 𝑆 = (𝑅 β†Ύs 𝐴)
issubdrg.z 0 = (0gβ€˜π‘…)
issubdrg.i 𝐼 = (invrβ€˜π‘…)
Assertion
Ref Expression
issubdrg ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ (𝑆 ∈ DivRing ↔ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑅   π‘₯,𝑆   π‘₯, 0
Allowed substitution hint:   𝐼(π‘₯)

Proof of Theorem issubdrg
StepHypRef Expression
1 simpllr 773 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ 𝐴 ∈ (SubRingβ€˜π‘…))
2 issubdrg.s . . . . . . 7 𝑆 = (𝑅 β†Ύs 𝐴)
32subrgring 20476 . . . . . 6 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
41, 3syl 17 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ 𝑆 ∈ Ring)
5 simpr 484 . . . . . . . . 9 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ ∈ (𝐴 βˆ– { 0 }))
6 eldifsn 4785 . . . . . . . . 9 (π‘₯ ∈ (𝐴 βˆ– { 0 }) ↔ (π‘₯ ∈ 𝐴 ∧ π‘₯ β‰  0 ))
75, 6sylib 217 . . . . . . . 8 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ (π‘₯ ∈ 𝐴 ∧ π‘₯ β‰  0 ))
87simpld 494 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ ∈ 𝐴)
92subrgbas 20483 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
101, 9syl 17 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ 𝐴 = (Baseβ€˜π‘†))
118, 10eleqtrd 2829 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
127simprd 495 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ β‰  0 )
13 issubdrg.z . . . . . . . . 9 0 = (0gβ€˜π‘…)
142, 13subrg0 20481 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 0 = (0gβ€˜π‘†))
151, 14syl 17 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ 0 = (0gβ€˜π‘†))
1612, 15neeqtrd 3004 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ β‰  (0gβ€˜π‘†))
17 eqid 2726 . . . . . . . 8 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
18 eqid 2726 . . . . . . . 8 (Unitβ€˜π‘†) = (Unitβ€˜π‘†)
19 eqid 2726 . . . . . . . 8 (0gβ€˜π‘†) = (0gβ€˜π‘†)
2017, 18, 19drngunit 20592 . . . . . . 7 (𝑆 ∈ DivRing β†’ (π‘₯ ∈ (Unitβ€˜π‘†) ↔ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))))
2120ad2antlr 724 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ (π‘₯ ∈ (Unitβ€˜π‘†) ↔ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))))
2211, 16, 21mpbir2and 710 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ ∈ (Unitβ€˜π‘†))
23 eqid 2726 . . . . . 6 (invrβ€˜π‘†) = (invrβ€˜π‘†)
2418, 23, 17ringinvcl 20294 . . . . 5 ((𝑆 ∈ Ring ∧ π‘₯ ∈ (Unitβ€˜π‘†)) β†’ ((invrβ€˜π‘†)β€˜π‘₯) ∈ (Baseβ€˜π‘†))
254, 22, 24syl2anc 583 . . . 4 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ ((invrβ€˜π‘†)β€˜π‘₯) ∈ (Baseβ€˜π‘†))
26 issubdrg.i . . . . . 6 𝐼 = (invrβ€˜π‘…)
272, 26, 18, 23subrginv 20490 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ (Unitβ€˜π‘†)) β†’ (πΌβ€˜π‘₯) = ((invrβ€˜π‘†)β€˜π‘₯))
281, 22, 27syl2anc 583 . . . 4 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ (πΌβ€˜π‘₯) = ((invrβ€˜π‘†)β€˜π‘₯))
2925, 28, 103eltr4d 2842 . . 3 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ (πΌβ€˜π‘₯) ∈ 𝐴)
3029ralrimiva 3140 . 2 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) β†’ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴)
313ad2antlr 724 . . 3 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 𝑆 ∈ Ring)
32 eqid 2726 . . . . . . . . . 10 (Unitβ€˜π‘…) = (Unitβ€˜π‘…)
332, 32, 18subrguss 20489 . . . . . . . . 9 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (Unitβ€˜π‘†) βŠ† (Unitβ€˜π‘…))
3433ad2antlr 724 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) βŠ† (Unitβ€˜π‘…))
35 eqid 2726 . . . . . . . . . . 11 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
3635, 32, 13isdrng 20591 . . . . . . . . . 10 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unitβ€˜π‘…) = ((Baseβ€˜π‘…) βˆ– { 0 })))
3736simprbi 496 . . . . . . . . 9 (𝑅 ∈ DivRing β†’ (Unitβ€˜π‘…) = ((Baseβ€˜π‘…) βˆ– { 0 }))
3837ad2antrr 723 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘…) = ((Baseβ€˜π‘…) βˆ– { 0 }))
3934, 38sseqtrd 4017 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) βŠ† ((Baseβ€˜π‘…) βˆ– { 0 }))
4017, 18unitss 20278 . . . . . . . 8 (Unitβ€˜π‘†) βŠ† (Baseβ€˜π‘†)
419ad2antlr 724 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 𝐴 = (Baseβ€˜π‘†))
4240, 41sseqtrrid 4030 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) βŠ† 𝐴)
4339, 42ssind 4227 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) βŠ† (((Baseβ€˜π‘…) βˆ– { 0 }) ∩ 𝐴))
4435subrgss 20474 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
4544ad2antlr 724 . . . . . . 7 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
46 difin2 4286 . . . . . . 7 (𝐴 βŠ† (Baseβ€˜π‘…) β†’ (𝐴 βˆ– { 0 }) = (((Baseβ€˜π‘…) βˆ– { 0 }) ∩ 𝐴))
4745, 46syl 17 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (𝐴 βˆ– { 0 }) = (((Baseβ€˜π‘…) βˆ– { 0 }) ∩ 𝐴))
4843, 47sseqtrrd 4018 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) βŠ† (𝐴 βˆ– { 0 }))
4944ad2antlr 724 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
50 simprl 768 . . . . . . . . . . . . . 14 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ ∈ (𝐴 βˆ– { 0 }))
5150, 6sylib 217 . . . . . . . . . . . . 13 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ (π‘₯ ∈ 𝐴 ∧ π‘₯ β‰  0 ))
5251simpld 494 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ ∈ 𝐴)
5349, 52sseldd 3978 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ ∈ (Baseβ€˜π‘…))
5451simprd 495 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ β‰  0 )
5535, 32, 13drngunit 20592 . . . . . . . . . . . 12 (𝑅 ∈ DivRing β†’ (π‘₯ ∈ (Unitβ€˜π‘…) ↔ (π‘₯ ∈ (Baseβ€˜π‘…) ∧ π‘₯ β‰  0 )))
5655ad2antrr 723 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ (π‘₯ ∈ (Unitβ€˜π‘…) ↔ (π‘₯ ∈ (Baseβ€˜π‘…) ∧ π‘₯ β‰  0 )))
5753, 54, 56mpbir2and 710 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ ∈ (Unitβ€˜π‘…))
58 simprr 770 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ (πΌβ€˜π‘₯) ∈ 𝐴)
592, 32, 18, 26subrgunit 20492 . . . . . . . . . . 11 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (π‘₯ ∈ (Unitβ€˜π‘†) ↔ (π‘₯ ∈ (Unitβ€˜π‘…) ∧ π‘₯ ∈ 𝐴 ∧ (πΌβ€˜π‘₯) ∈ 𝐴)))
6059ad2antlr 724 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ (π‘₯ ∈ (Unitβ€˜π‘†) ↔ (π‘₯ ∈ (Unitβ€˜π‘…) ∧ π‘₯ ∈ 𝐴 ∧ (πΌβ€˜π‘₯) ∈ 𝐴)))
6157, 52, 58, 60mpbir3and 1339 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ ∈ (Unitβ€˜π‘†))
6261expr 456 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ ((πΌβ€˜π‘₯) ∈ 𝐴 β†’ π‘₯ ∈ (Unitβ€˜π‘†)))
6362ralimdva 3161 . . . . . . 7 ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ (βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴 β†’ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })π‘₯ ∈ (Unitβ€˜π‘†)))
6463imp 406 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })π‘₯ ∈ (Unitβ€˜π‘†))
65 dfss3 3965 . . . . . 6 ((𝐴 βˆ– { 0 }) βŠ† (Unitβ€˜π‘†) ↔ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })π‘₯ ∈ (Unitβ€˜π‘†))
6664, 65sylibr 233 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (𝐴 βˆ– { 0 }) βŠ† (Unitβ€˜π‘†))
6748, 66eqssd 3994 . . . 4 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) = (𝐴 βˆ– { 0 }))
6814ad2antlr 724 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 0 = (0gβ€˜π‘†))
6968sneqd 4635 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ { 0 } = {(0gβ€˜π‘†)})
7041, 69difeq12d 4118 . . . 4 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (𝐴 βˆ– { 0 }) = ((Baseβ€˜π‘†) βˆ– {(0gβ€˜π‘†)}))
7167, 70eqtrd 2766 . . 3 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) = ((Baseβ€˜π‘†) βˆ– {(0gβ€˜π‘†)}))
7217, 18, 19isdrng 20591 . . 3 (𝑆 ∈ DivRing ↔ (𝑆 ∈ Ring ∧ (Unitβ€˜π‘†) = ((Baseβ€˜π‘†) βˆ– {(0gβ€˜π‘†)})))
7331, 71, 72sylanbrc 582 . 2 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 𝑆 ∈ DivRing)
7430, 73impbida 798 1 ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ (𝑆 ∈ DivRing ↔ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055   βˆ– cdif 3940   ∩ cin 3942   βŠ† wss 3943  {csn 4623  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153   β†Ύs cress 17182  0gc0g 17394  Ringcrg 20138  Unitcui 20257  invrcinvr 20289  SubRingcsubrg 20469  DivRingcdr 20587
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-subg 19050  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-oppr 20236  df-dvdsr 20259  df-unit 20260  df-invr 20290  df-subrg 20471  df-drng 20589
This theorem is referenced by:  issdrg2  20646  cnsubdrglem  21312  extdg1id  33260
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