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Theorem isdrng4 32822
Description: A division ring is a ring in which 1 β‰  0 and every nonzero element has a left and right inverse. (Contributed by Thierry Arnoux, 2-Mar-2025.)
Hypotheses
Ref Expression
isdrng4.b 𝐡 = (Baseβ€˜π‘…)
isdrng4.0 0 = (0gβ€˜π‘…)
isdrng4.1 1 = (1rβ€˜π‘…)
isdrng4.x Β· = (.rβ€˜π‘…)
isdrng4.u π‘ˆ = (Unitβ€˜π‘…)
isdrng4.r (πœ‘ β†’ 𝑅 ∈ Ring)
Assertion
Ref Expression
isdrng4 (πœ‘ β†’ (𝑅 ∈ DivRing ↔ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))))
Distinct variable groups:   π‘₯, 0   π‘₯, 1 ,𝑦   π‘₯, Β· ,𝑦   π‘₯,𝐡,𝑦   π‘₯,𝑅,𝑦   π‘₯,π‘ˆ,𝑦   πœ‘,π‘₯,𝑦
Allowed substitution hint:   0 (𝑦)
Proof of Theorem isdrng4
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 isdrng4.b . . . 4 𝐡 = (Baseβ€˜π‘…)
2 isdrng4.u . . . 4 π‘ˆ = (Unitβ€˜π‘…)
3 isdrng4.0 . . . 4 0 = (0gβ€˜π‘…)
41, 2, 3isdrng 20576 . . 3 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ π‘ˆ = (𝐡 βˆ– { 0 })))
5 isdrng4.r . . . 4 (πœ‘ β†’ 𝑅 ∈ Ring)
65biantrurd 532 . . 3 (πœ‘ β†’ (π‘ˆ = (𝐡 βˆ– { 0 }) ↔ (𝑅 ∈ Ring ∧ π‘ˆ = (𝐡 βˆ– { 0 }))))
74, 6bitr4id 290 . 2 (πœ‘ β†’ (𝑅 ∈ DivRing ↔ π‘ˆ = (𝐡 βˆ– { 0 })))
8 isdrng4.1 . . . . . . . . 9 1 = (1rβ€˜π‘…)
92, 81unit 20261 . . . . . . . 8 (𝑅 ∈ Ring β†’ 1 ∈ π‘ˆ)
105, 9syl 17 . . . . . . 7 (πœ‘ β†’ 1 ∈ π‘ˆ)
1110adantr 480 . . . . . 6 ((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) β†’ 1 ∈ π‘ˆ)
12 simpr 484 . . . . . 6 ((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) β†’ π‘ˆ = (𝐡 βˆ– { 0 }))
1311, 12eleqtrd 2827 . . . . 5 ((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) β†’ 1 ∈ (𝐡 βˆ– { 0 }))
14 eldifsni 4785 . . . . 5 ( 1 ∈ (𝐡 βˆ– { 0 }) β†’ 1 β‰  0 )
1513, 14syl 17 . . . 4 ((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) β†’ 1 β‰  0 )
16 simpll 764 . . . . . 6 (((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) β†’ πœ‘)
1712eleq2d 2811 . . . . . . 7 ((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) β†’ (π‘₯ ∈ π‘ˆ ↔ π‘₯ ∈ (𝐡 βˆ– { 0 })))
1817biimpar 477 . . . . . 6 (((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) β†’ π‘₯ ∈ π‘ˆ)
19 isdrng4.x . . . . . . . . . . . . 13 Β· = (.rβ€˜π‘…)
205ad5antr 731 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ 𝑅 ∈ Ring)
211, 2unitcl 20262 . . . . . . . . . . . . . 14 (π‘₯ ∈ π‘ˆ β†’ π‘₯ ∈ 𝐡)
2221ad5antlr 732 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ π‘₯ ∈ 𝐡)
23 simp-4r 781 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ 𝑦 ∈ 𝐡)
24 simplr 766 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ 𝑧 ∈ 𝐡)
25 simpllr 773 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ (𝑦 Β· π‘₯) = 1 )
26 simpr 484 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ (π‘₯ Β· 𝑧) = 1 )
271, 3, 8, 19, 2, 20, 22, 23, 24, 25, 26ringinveu 32821 . . . . . . . . . . . 12 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ 𝑧 = 𝑦)
2827oveq2d 7417 . . . . . . . . . . 11 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ (π‘₯ Β· 𝑧) = (π‘₯ Β· 𝑦))
2928, 26eqtr3d 2766 . . . . . . . . . 10 ((((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) ∧ 𝑧 ∈ 𝐡) ∧ (π‘₯ Β· 𝑧) = 1 ) β†’ (π‘₯ Β· 𝑦) = 1 )
3021ad3antlr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) β†’ π‘₯ ∈ 𝐡)
31 eqid 2724 . . . . . . . . . . . . . 14 (βˆ₯rβ€˜π‘…) = (βˆ₯rβ€˜π‘…)
32 eqid 2724 . . . . . . . . . . . . . 14 (opprβ€˜π‘…) = (opprβ€˜π‘…)
33 eqid 2724 . . . . . . . . . . . . . 14 (βˆ₯rβ€˜(opprβ€˜π‘…)) = (βˆ₯rβ€˜(opprβ€˜π‘…))
342, 8, 31, 32, 33isunit 20260 . . . . . . . . . . . . 13 (π‘₯ ∈ π‘ˆ ↔ (π‘₯(βˆ₯rβ€˜π‘…) 1 ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 ))
3534simprbi 496 . . . . . . . . . . . 12 (π‘₯ ∈ π‘ˆ β†’ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 )
3635ad3antlr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) β†’ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 )
3732, 1opprbas 20228 . . . . . . . . . . . . . . 15 𝐡 = (Baseβ€˜(opprβ€˜π‘…))
38 eqid 2724 . . . . . . . . . . . . . . 15 (.rβ€˜(opprβ€˜π‘…)) = (.rβ€˜(opprβ€˜π‘…))
3937, 33, 38dvdsr2 20250 . . . . . . . . . . . . . 14 (π‘₯ ∈ 𝐡 β†’ (π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 ↔ βˆƒπ‘¦ ∈ 𝐡 (𝑦(.rβ€˜(opprβ€˜π‘…))π‘₯) = 1 ))
4039biimpa 476 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐡 ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 ) β†’ βˆƒπ‘¦ ∈ 𝐡 (𝑦(.rβ€˜(opprβ€˜π‘…))π‘₯) = 1 )
411, 19, 32, 38opprmul 20224 . . . . . . . . . . . . . . 15 (𝑦(.rβ€˜(opprβ€˜π‘…))π‘₯) = (π‘₯ Β· 𝑦)
4241eqeq1i 2729 . . . . . . . . . . . . . 14 ((𝑦(.rβ€˜(opprβ€˜π‘…))π‘₯) = 1 ↔ (π‘₯ Β· 𝑦) = 1 )
4342rexbii 3086 . . . . . . . . . . . . 13 (βˆƒπ‘¦ ∈ 𝐡 (𝑦(.rβ€˜(opprβ€˜π‘…))π‘₯) = 1 ↔ βˆƒπ‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) = 1 )
4440, 43sylib 217 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝐡 ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 ) β†’ βˆƒπ‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) = 1 )
45 oveq2 7409 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 β†’ (π‘₯ Β· 𝑦) = (π‘₯ Β· 𝑧))
4645eqeq1d 2726 . . . . . . . . . . . . 13 (𝑦 = 𝑧 β†’ ((π‘₯ Β· 𝑦) = 1 ↔ (π‘₯ Β· 𝑧) = 1 ))
4746cbvrexvw 3227 . . . . . . . . . . . 12 (βˆƒπ‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) = 1 ↔ βˆƒπ‘§ ∈ 𝐡 (π‘₯ Β· 𝑧) = 1 )
4844, 47sylib 217 . . . . . . . . . . 11 ((π‘₯ ∈ 𝐡 ∧ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 ) β†’ βˆƒπ‘§ ∈ 𝐡 (π‘₯ Β· 𝑧) = 1 )
4930, 36, 48syl2anc 583 . . . . . . . . . 10 ((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) β†’ βˆƒπ‘§ ∈ 𝐡 (π‘₯ Β· 𝑧) = 1 )
5029, 49r19.29a 3154 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) β†’ (π‘₯ Β· 𝑦) = 1 )
51 simpr 484 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) β†’ (𝑦 Β· π‘₯) = 1 )
5250, 51jca 511 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ 𝑦 ∈ 𝐡) ∧ (𝑦 Β· π‘₯) = 1 ) β†’ ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))
5352anasss 466 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ π‘ˆ) ∧ (𝑦 ∈ 𝐡 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))
5421adantl 481 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ π‘₯ ∈ 𝐡)
5534simplbi 497 . . . . . . . . 9 (π‘₯ ∈ π‘ˆ β†’ π‘₯(βˆ₯rβ€˜π‘…) 1 )
5655adantl 481 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ π‘₯(βˆ₯rβ€˜π‘…) 1 )
571, 31, 19dvdsr2 20250 . . . . . . . . 9 (π‘₯ ∈ 𝐡 β†’ (π‘₯(βˆ₯rβ€˜π‘…) 1 ↔ βˆƒπ‘¦ ∈ 𝐡 (𝑦 Β· π‘₯) = 1 ))
5857biimpa 476 . . . . . . . 8 ((π‘₯ ∈ 𝐡 ∧ π‘₯(βˆ₯rβ€˜π‘…) 1 ) β†’ βˆƒπ‘¦ ∈ 𝐡 (𝑦 Β· π‘₯) = 1 )
5954, 56, 58syl2anc 583 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ βˆƒπ‘¦ ∈ 𝐡 (𝑦 Β· π‘₯) = 1 )
6053, 59reximddv 3163 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))
6116, 18, 60syl2anc 583 . . . . 5 (((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) β†’ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))
6261ralrimiva 3138 . . . 4 ((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) β†’ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))
6315, 62jca 511 . . 3 ((πœ‘ ∧ π‘ˆ = (𝐡 βˆ– { 0 })) β†’ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )))
641, 2unitss 20263 . . . . . 6 π‘ˆ βŠ† 𝐡
6564a1i 11 . . . . 5 ((πœ‘ ∧ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))) β†’ π‘ˆ βŠ† 𝐡)
665adantr 480 . . . . . 6 ((πœ‘ ∧ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))) β†’ 𝑅 ∈ Ring)
67 simprl 768 . . . . . 6 ((πœ‘ ∧ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))) β†’ 1 β‰  0 )
682, 3, 80unit 20283 . . . . . . . 8 (𝑅 ∈ Ring β†’ ( 0 ∈ π‘ˆ ↔ 1 = 0 ))
6968necon3bbid 2970 . . . . . . 7 (𝑅 ∈ Ring β†’ (Β¬ 0 ∈ π‘ˆ ↔ 1 β‰  0 ))
7069biimpar 477 . . . . . 6 ((𝑅 ∈ Ring ∧ 1 β‰  0 ) β†’ Β¬ 0 ∈ π‘ˆ)
7166, 67, 70syl2anc 583 . . . . 5 ((πœ‘ ∧ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))) β†’ Β¬ 0 ∈ π‘ˆ)
72 ssdifsn 4783 . . . . 5 (π‘ˆ βŠ† (𝐡 βˆ– { 0 }) ↔ (π‘ˆ βŠ† 𝐡 ∧ Β¬ 0 ∈ π‘ˆ))
7365, 71, 72sylanbrc 582 . . . 4 ((πœ‘ ∧ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))) β†’ π‘ˆ βŠ† (𝐡 βˆ– { 0 }))
74 simplr 766 . . . . . . . . . . 11 ((((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) ∧ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ π‘₯ ∈ (𝐡 βˆ– { 0 }))
7574eldifad 3952 . . . . . . . . . 10 ((((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) ∧ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ π‘₯ ∈ 𝐡)
76 simpr 484 . . . . . . . . . . . 12 (((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ) β†’ (𝑦 Β· π‘₯) = 1 )
7776reximi 3076 . . . . . . . . . . 11 (βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ) β†’ βˆƒπ‘¦ ∈ 𝐡 (𝑦 Β· π‘₯) = 1 )
7877adantl 481 . . . . . . . . . 10 ((((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) ∧ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ βˆƒπ‘¦ ∈ 𝐡 (𝑦 Β· π‘₯) = 1 )
7957biimpar 477 . . . . . . . . . 10 ((π‘₯ ∈ 𝐡 ∧ βˆƒπ‘¦ ∈ 𝐡 (𝑦 Β· π‘₯) = 1 ) β†’ π‘₯(βˆ₯rβ€˜π‘…) 1 )
8075, 78, 79syl2anc 583 . . . . . . . . 9 ((((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) ∧ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ π‘₯(βˆ₯rβ€˜π‘…) 1 )
81 simpl 482 . . . . . . . . . . . . 13 (((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ) β†’ (π‘₯ Β· 𝑦) = 1 )
8281reximi 3076 . . . . . . . . . . . 12 (βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ) β†’ βˆƒπ‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) = 1 )
8382adantl 481 . . . . . . . . . . 11 ((((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) ∧ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ βˆƒπ‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) = 1 )
8483, 43sylibr 233 . . . . . . . . . 10 ((((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) ∧ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ βˆƒπ‘¦ ∈ 𝐡 (𝑦(.rβ€˜(opprβ€˜π‘…))π‘₯) = 1 )
8539biimpar 477 . . . . . . . . . 10 ((π‘₯ ∈ 𝐡 ∧ βˆƒπ‘¦ ∈ 𝐡 (𝑦(.rβ€˜(opprβ€˜π‘…))π‘₯) = 1 ) β†’ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 )
8675, 84, 85syl2anc 583 . . . . . . . . 9 ((((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) ∧ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ π‘₯(βˆ₯rβ€˜(opprβ€˜π‘…)) 1 )
8780, 86, 34sylanbrc 582 . . . . . . . 8 ((((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) ∧ βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 )) β†’ π‘₯ ∈ π‘ˆ)
8887ex 412 . . . . . . 7 (((πœ‘ ∧ 1 β‰  0 ) ∧ π‘₯ ∈ (𝐡 βˆ– { 0 })) β†’ (βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ) β†’ π‘₯ ∈ π‘ˆ))
8988ralimdva 3159 . . . . . 6 ((πœ‘ ∧ 1 β‰  0 ) β†’ (βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ) β†’ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })π‘₯ ∈ π‘ˆ))
9089impr 454 . . . . 5 ((πœ‘ ∧ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))) β†’ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })π‘₯ ∈ π‘ˆ)
91 dfss3 3962 . . . . 5 ((𝐡 βˆ– { 0 }) βŠ† π‘ˆ ↔ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })π‘₯ ∈ π‘ˆ)
9290, 91sylibr 233 . . . 4 ((πœ‘ ∧ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))) β†’ (𝐡 βˆ– { 0 }) βŠ† π‘ˆ)
9373, 92eqssd 3991 . . 3 ((πœ‘ ∧ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))) β†’ π‘ˆ = (𝐡 βˆ– { 0 }))
9463, 93impbida 798 . 2 (πœ‘ β†’ (π‘ˆ = (𝐡 βˆ– { 0 }) ↔ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))))
957, 94bitrd 279 1 (πœ‘ β†’ (𝑅 ∈ DivRing ↔ ( 1 β‰  0 ∧ βˆ€π‘₯ ∈ (𝐡 βˆ– { 0 })βˆƒπ‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 1 ∧ (𝑦 Β· π‘₯) = 1 ))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053  βˆƒwrex 3062   βˆ– cdif 3937   βŠ† wss 3940  {csn 4620   class class class wbr 5138  β€˜cfv 6533  (class class class)co 7401  Basecbs 17140  .rcmulr 17194  0gc0g 17381  1rcur 20071  Ringcrg 20123  opprcoppr 20220  βˆ₯rcdsr 20241  Unitcui 20242  DivRingcdr 20572
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 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-0g 17383  df-mgm 18560  df-sgrp 18639  df-mnd 18655  df-grp 18853  df-minusg 18854  df-cmn 19687  df-abl 19688  df-mgp 20025  df-rng 20043  df-ur 20072  df-ring 20125  df-oppr 20221  df-dvdsr 20244  df-unit 20245  df-invr 20275  df-drng 20574
This theorem is referenced by:  drngidl  32982  opprqusdrng  33038  qsdrngi  33040
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