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Theorem issubdrg 20672
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 774 . . . . . 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 20517 . . . . . 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 483 . . . . . . . . 9 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ ∈ (𝐴 βˆ– { 0 }))
6 eldifsn 4786 . . . . . . . . 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 493 . . . . . . 7 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ ∈ 𝐴)
92subrgbas 20524 . . . . . . . 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 2827 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
127simprd 494 . . . . . . 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 20522 . . . . . . . 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 3000 . . . . . 6 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ β‰  (0gβ€˜π‘†))
17 eqid 2725 . . . . . . . 8 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
18 eqid 2725 . . . . . . . 8 (Unitβ€˜π‘†) = (Unitβ€˜π‘†)
19 eqid 2725 . . . . . . . 8 (0gβ€˜π‘†) = (0gβ€˜π‘†)
2017, 18, 19drngunit 20633 . . . . . . 7 (𝑆 ∈ DivRing β†’ (π‘₯ ∈ (Unitβ€˜π‘†) ↔ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ π‘₯ β‰  (0gβ€˜π‘†))))
2120ad2antlr 725 . . . . . 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 711 . . . . 5 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ π‘₯ ∈ (Unitβ€˜π‘†))
23 eqid 2725 . . . . . 6 (invrβ€˜π‘†) = (invrβ€˜π‘†)
2418, 23, 17ringinvcl 20335 . . . . 5 ((𝑆 ∈ Ring ∧ π‘₯ ∈ (Unitβ€˜π‘†)) β†’ ((invrβ€˜π‘†)β€˜π‘₯) ∈ (Baseβ€˜π‘†))
254, 22, 24syl2anc 582 . . . 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 20531 . . . . 5 ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ (Unitβ€˜π‘†)) β†’ (πΌβ€˜π‘₯) = ((invrβ€˜π‘†)β€˜π‘₯))
281, 22, 27syl2anc 582 . . . 4 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ (πΌβ€˜π‘₯) = ((invrβ€˜π‘†)β€˜π‘₯))
2925, 28, 103eltr4d 2840 . . 3 ((((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ (πΌβ€˜π‘₯) ∈ 𝐴)
3029ralrimiva 3136 . 2 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ 𝑆 ∈ DivRing) β†’ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴)
313ad2antlr 725 . . 3 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 𝑆 ∈ Ring)
32 eqid 2725 . . . . . . . . . 10 (Unitβ€˜π‘…) = (Unitβ€˜π‘…)
332, 32, 18subrguss 20530 . . . . . . . . 9 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (Unitβ€˜π‘†) βŠ† (Unitβ€˜π‘…))
3433ad2antlr 725 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) βŠ† (Unitβ€˜π‘…))
35 eqid 2725 . . . . . . . . . . 11 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
3635, 32, 13isdrng 20632 . . . . . . . . . 10 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unitβ€˜π‘…) = ((Baseβ€˜π‘…) βˆ– { 0 })))
3736simprbi 495 . . . . . . . . 9 (𝑅 ∈ DivRing β†’ (Unitβ€˜π‘…) = ((Baseβ€˜π‘…) βˆ– { 0 }))
3837ad2antrr 724 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘…) = ((Baseβ€˜π‘…) βˆ– { 0 }))
3934, 38sseqtrd 4013 . . . . . . 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 20319 . . . . . . . 8 (Unitβ€˜π‘†) βŠ† (Baseβ€˜π‘†)
419ad2antlr 725 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 𝐴 = (Baseβ€˜π‘†))
4240, 41sseqtrrid 4026 . . . . . . 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 20515 . . . . . . . 8 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
4544ad2antlr 725 . . . . . . 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 4014 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) βŠ† (𝐴 βˆ– { 0 }))
4944ad2antlr 725 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
50 simprl 769 . . . . . . . . . . . . . 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 493 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ ∈ 𝐴)
5349, 52sseldd 3973 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ ∈ (Baseβ€˜π‘…))
5451simprd 494 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ β‰  0 )
5535, 32, 13drngunit 20633 . . . . . . . . . . . 12 (𝑅 ∈ DivRing β†’ (π‘₯ ∈ (Unitβ€˜π‘…) ↔ (π‘₯ ∈ (Baseβ€˜π‘…) ∧ π‘₯ β‰  0 )))
5655ad2antrr 724 . . . . . . . . . . 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 711 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ π‘₯ ∈ (Unitβ€˜π‘…))
58 simprr 771 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ (𝐴 βˆ– { 0 }) ∧ (πΌβ€˜π‘₯) ∈ 𝐴)) β†’ (πΌβ€˜π‘₯) ∈ 𝐴)
592, 32, 18, 26subrgunit 20533 . . . . . . . . . . 11 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (π‘₯ ∈ (Unitβ€˜π‘†) ↔ (π‘₯ ∈ (Unitβ€˜π‘…) ∧ π‘₯ ∈ 𝐴 ∧ (πΌβ€˜π‘₯) ∈ 𝐴)))
6059ad2antlr 725 . . . . . . . . . 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 455 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ (𝐴 βˆ– { 0 })) β†’ ((πΌβ€˜π‘₯) ∈ 𝐴 β†’ π‘₯ ∈ (Unitβ€˜π‘†)))
6362ralimdva 3157 . . . . . . 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 405 . . . . . 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 3960 . . . . . 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 3990 . . . 4 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ (Unitβ€˜π‘†) = (𝐴 βˆ– { 0 }))
6814ad2antlr 725 . . . . . 6 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 0 = (0gβ€˜π‘†))
6968sneqd 4636 . . . . 5 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ { 0 } = {(0gβ€˜π‘†)})
7041, 69difeq12d 4115 . . . 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 2765 . . 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 20632 . . 3 (𝑆 ∈ DivRing ↔ (𝑆 ∈ Ring ∧ (Unitβ€˜π‘†) = ((Baseβ€˜π‘†) βˆ– {(0gβ€˜π‘†)})))
7331, 71, 72sylanbrc 581 . 2 (((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) ∧ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴) β†’ 𝑆 ∈ DivRing)
7430, 73impbida 799 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051   βˆ– cdif 3936   ∩ cin 3938   βŠ† wss 3939  {csn 4624  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179   β†Ύs cress 17208  0gc0g 17420  Ringcrg 20177  Unitcui 20298  invrcinvr 20330  SubRingcsubrg 20510  DivRingcdr 20628
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7992  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18897  df-minusg 18898  df-subg 19082  df-cmn 19741  df-abl 19742  df-mgp 20079  df-rng 20097  df-ur 20126  df-ring 20179  df-oppr 20277  df-dvdsr 20300  df-unit 20301  df-invr 20331  df-subrg 20512  df-drng 20630
This theorem is referenced by:  issdrg2  20687  cnsubdrglem  21355  extdg1id  33412
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