Step | Hyp | Ref
| Expression |
1 | | simpr 486 |
. 2
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β (π
βΎs π΄) β Ring) |
2 | | ringgrp 19829 |
. . . 4
β’ ((π
βΎs π΄) β Ring β (π
βΎs π΄) β Grp) |
3 | 2 | adantl 483 |
. . 3
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β (π
βΎs π΄) β Grp) |
4 | | orngogrp 31541 |
. . . . 5
β’ (π
β oRing β π
β oGrp) |
5 | | isogrp 31369 |
. . . . . 6
β’ (π
β oGrp β (π
β Grp β§ π
β oMnd)) |
6 | 5 | simprbi 498 |
. . . . 5
β’ (π
β oGrp β π
β oMnd) |
7 | 4, 6 | syl 17 |
. . . 4
β’ (π
β oRing β π
β oMnd) |
8 | | ringmnd 19834 |
. . . 4
β’ ((π
βΎs π΄) β Ring β (π
βΎs π΄) β Mnd) |
9 | | submomnd 31377 |
. . . 4
β’ ((π
β oMnd β§ (π
βΎs π΄) β Mnd) β (π
βΎs π΄) β oMnd) |
10 | 7, 8, 9 | syl2an 597 |
. . 3
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β (π
βΎs π΄) β oMnd) |
11 | | isogrp 31369 |
. . 3
β’ ((π
βΎs π΄) β oGrp β ((π
βΎs π΄) β Grp β§ (π
βΎs π΄) β oMnd)) |
12 | 3, 10, 11 | sylanbrc 584 |
. 2
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β (π
βΎs π΄) β oGrp) |
13 | | simp-4l 781 |
. . . . . . 7
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β π
β oRing) |
14 | | reldmress 16984 |
. . . . . . . . . . . . . . 15
β’ Rel dom
βΎs |
15 | 14 | ovprc2 7343 |
. . . . . . . . . . . . . 14
β’ (Β¬
π΄ β V β (π
βΎs π΄) = β
) |
16 | 15 | fveq2d 6804 |
. . . . . . . . . . . . 13
β’ (Β¬
π΄ β V β
(Baseβ(π
βΎs π΄)) =
(Baseββ
)) |
17 | 16 | adantl 483 |
. . . . . . . . . . . 12
β’ (((π
β oRing β§ (π
βΎs π΄) β Ring) β§ Β¬ π΄ β V) β
(Baseβ(π
βΎs π΄)) =
(Baseββ
)) |
18 | | base0 16958 |
. . . . . . . . . . . 12
β’ β
=
(Baseββ
) |
19 | 17, 18 | eqtr4di 2794 |
. . . . . . . . . . 11
β’ (((π
β oRing β§ (π
βΎs π΄) β Ring) β§ Β¬ π΄ β V) β
(Baseβ(π
βΎs π΄)) =
β
) |
20 | | eqid 2736 |
. . . . . . . . . . . . . . 15
β’
(Baseβ(π
βΎs π΄)) =
(Baseβ(π
βΎs π΄)) |
21 | | eqid 2736 |
. . . . . . . . . . . . . . 15
β’
(1rβ(π
βΎs π΄)) = (1rβ(π
βΎs π΄)) |
22 | 20, 21 | ringidcl 19848 |
. . . . . . . . . . . . . 14
β’ ((π
βΎs π΄) β Ring β
(1rβ(π
βΎs π΄))
β (Baseβ(π
βΎs π΄))) |
23 | 22 | ne0d 4275 |
. . . . . . . . . . . . 13
β’ ((π
βΎs π΄) β Ring β
(Baseβ(π
βΎs π΄))
β β
) |
24 | 23 | ad2antlr 725 |
. . . . . . . . . . . 12
β’ (((π
β oRing β§ (π
βΎs π΄) β Ring) β§ Β¬ π΄ β V) β
(Baseβ(π
βΎs π΄))
β β
) |
25 | 24 | neneqd 2946 |
. . . . . . . . . . 11
β’ (((π
β oRing β§ (π
βΎs π΄) β Ring) β§ Β¬ π΄ β V) β Β¬
(Baseβ(π
βΎs π΄)) =
β
) |
26 | 19, 25 | condan 816 |
. . . . . . . . . 10
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β π΄ β V) |
27 | | eqid 2736 |
. . . . . . . . . . . 12
β’ (π
βΎs π΄) = (π
βΎs π΄) |
28 | | eqid 2736 |
. . . . . . . . . . . 12
β’
(Baseβπ
) =
(Baseβπ
) |
29 | 27, 28 | ressbas 16988 |
. . . . . . . . . . 11
β’ (π΄ β V β (π΄ β© (Baseβπ
)) = (Baseβ(π
βΎs π΄))) |
30 | | inss2 4169 |
. . . . . . . . . . 11
β’ (π΄ β© (Baseβπ
)) β (Baseβπ
) |
31 | 29, 30 | eqsstrrdi 3981 |
. . . . . . . . . 10
β’ (π΄ β V β
(Baseβ(π
βΎs π΄))
β (Baseβπ
)) |
32 | 26, 31 | syl 17 |
. . . . . . . . 9
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β
(Baseβ(π
βΎs π΄))
β (Baseβπ
)) |
33 | 32 | ad3antrrr 728 |
. . . . . . . 8
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (Baseβ(π
βΎs π΄)) β (Baseβπ
)) |
34 | | simpllr 774 |
. . . . . . . 8
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β π β (Baseβ(π
βΎs π΄))) |
35 | 33, 34 | sseldd 3927 |
. . . . . . 7
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β π β (Baseβπ
)) |
36 | | simprl 769 |
. . . . . . . 8
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π) |
37 | | orngring 31540 |
. . . . . . . . . . . . . . . 16
β’ (π
β oRing β π
β Ring) |
38 | | ringgrp 19829 |
. . . . . . . . . . . . . . . 16
β’ (π
β Ring β π
β Grp) |
39 | 37, 38 | syl 17 |
. . . . . . . . . . . . . . 15
β’ (π
β oRing β π
β Grp) |
40 | 39 | adantr 482 |
. . . . . . . . . . . . . 14
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β π
β Grp) |
41 | 28 | ressinbas 16996 |
. . . . . . . . . . . . . . . . 17
β’ (π΄ β V β (π
βΎs π΄) = (π
βΎs (π΄ β© (Baseβπ
)))) |
42 | 29 | oveq2d 7319 |
. . . . . . . . . . . . . . . . 17
β’ (π΄ β V β (π
βΎs (π΄ β© (Baseβπ
))) = (π
βΎs (Baseβ(π
βΎs π΄)))) |
43 | 41, 42 | eqtrd 2776 |
. . . . . . . . . . . . . . . 16
β’ (π΄ β V β (π
βΎs π΄) = (π
βΎs (Baseβ(π
βΎs π΄)))) |
44 | 26, 43 | syl 17 |
. . . . . . . . . . . . . . 15
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β (π
βΎs π΄) = (π
βΎs (Baseβ(π
βΎs π΄)))) |
45 | 44, 3 | eqeltrrd 2838 |
. . . . . . . . . . . . . 14
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β (π
βΎs
(Baseβ(π
βΎs π΄)))
β Grp) |
46 | 28 | issubg 18796 |
. . . . . . . . . . . . . 14
β’
((Baseβ(π
βΎs π΄))
β (SubGrpβπ
)
β (π
β Grp β§
(Baseβ(π
βΎs π΄))
β (Baseβπ
)
β§ (π
βΎs (Baseβ(π
βΎs π΄))) β Grp)) |
47 | 40, 32, 45, 46 | syl3anbrc 1343 |
. . . . . . . . . . . . 13
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β
(Baseβ(π
βΎs π΄))
β (SubGrpβπ
)) |
48 | | eqid 2736 |
. . . . . . . . . . . . . 14
β’ (π
βΎs
(Baseβ(π
βΎs π΄))) =
(π
βΎs
(Baseβ(π
βΎs π΄))) |
49 | | eqid 2736 |
. . . . . . . . . . . . . 14
β’
(0gβπ
) = (0gβπ
) |
50 | 48, 49 | subg0 18802 |
. . . . . . . . . . . . 13
β’
((Baseβ(π
βΎs π΄))
β (SubGrpβπ
)
β (0gβπ
) = (0gβ(π
βΎs (Baseβ(π
βΎs π΄))))) |
51 | 47, 50 | syl 17 |
. . . . . . . . . . . 12
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β
(0gβπ
) =
(0gβ(π
βΎs (Baseβ(π
βΎs π΄))))) |
52 | 44 | fveq2d 6804 |
. . . . . . . . . . . 12
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β
(0gβ(π
βΎs π΄)) =
(0gβ(π
βΎs (Baseβ(π
βΎs π΄))))) |
53 | 51, 52 | eqtr4d 2779 |
. . . . . . . . . . 11
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β
(0gβπ
) =
(0gβ(π
βΎs π΄))) |
54 | 53 | ad2antrr 724 |
. . . . . . . . . 10
β’ ((((π
β oRing β§ (π
βΎs π΄) β Ring) β§ π β (Baseβ(π
βΎs π΄))) β§ π β (Baseβ(π
βΎs π΄))) β (0gβπ
) = (0gβ(π
βΎs π΄))) |
55 | 26 | ad2antrr 724 |
. . . . . . . . . . 11
β’ ((((π
β oRing β§ (π
βΎs π΄) β Ring) β§ π β (Baseβ(π
βΎs π΄))) β§ π β (Baseβ(π
βΎs π΄))) β π΄ β V) |
56 | | eqid 2736 |
. . . . . . . . . . . 12
β’
(leβπ
) =
(leβπ
) |
57 | 27, 56 | ressle 17131 |
. . . . . . . . . . 11
β’ (π΄ β V β (leβπ
) = (leβ(π
βΎs π΄))) |
58 | 55, 57 | syl 17 |
. . . . . . . . . 10
β’ ((((π
β oRing β§ (π
βΎs π΄) β Ring) β§ π β (Baseβ(π
βΎs π΄))) β§ π β (Baseβ(π
βΎs π΄))) β (leβπ
) = (leβ(π
βΎs π΄))) |
59 | | eqidd 2737 |
. . . . . . . . . 10
β’ ((((π
β oRing β§ (π
βΎs π΄) β Ring) β§ π β (Baseβ(π
βΎs π΄))) β§ π β (Baseβ(π
βΎs π΄))) β π = π) |
60 | 54, 58, 59 | breq123d 5095 |
. . . . . . . . 9
β’ ((((π
β oRing β§ (π
βΎs π΄) β Ring) β§ π β (Baseβ(π
βΎs π΄))) β§ π β (Baseβ(π
βΎs π΄))) β ((0gβπ
)(leβπ
)π β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) |
61 | 60 | adantr 482 |
. . . . . . . 8
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β ((0gβπ
)(leβπ
)π β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) |
62 | 36, 61 | mpbird 258 |
. . . . . . 7
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (0gβπ
)(leβπ
)π) |
63 | | simplr 767 |
. . . . . . . 8
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β π β (Baseβ(π
βΎs π΄))) |
64 | 33, 63 | sseldd 3927 |
. . . . . . 7
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β π β (Baseβπ
)) |
65 | | simprr 771 |
. . . . . . . 8
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π) |
66 | | eqidd 2737 |
. . . . . . . . . 10
β’ ((((π
β oRing β§ (π
βΎs π΄) β Ring) β§ π β (Baseβ(π
βΎs π΄))) β§ π β (Baseβ(π
βΎs π΄))) β π = π) |
67 | 54, 58, 66 | breq123d 5095 |
. . . . . . . . 9
β’ ((((π
β oRing β§ (π
βΎs π΄) β Ring) β§ π β (Baseβ(π
βΎs π΄))) β§ π β (Baseβ(π
βΎs π΄))) β ((0gβπ
)(leβπ
)π β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) |
68 | 67 | adantr 482 |
. . . . . . . 8
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β ((0gβπ
)(leβπ
)π β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) |
69 | 65, 68 | mpbird 258 |
. . . . . . 7
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (0gβπ
)(leβπ
)π) |
70 | | eqid 2736 |
. . . . . . . 8
β’
(.rβπ
) = (.rβπ
) |
71 | 28, 56, 49, 70 | orngmul 31543 |
. . . . . . 7
β’ ((π
β oRing β§ (π β (Baseβπ
) β§
(0gβπ
)(leβπ
)π) β§ (π β (Baseβπ
) β§ (0gβπ
)(leβπ
)π)) β (0gβπ
)(leβπ
)(π(.rβπ
)π)) |
72 | 13, 35, 62, 64, 69, 71 | syl122anc 1379 |
. . . . . 6
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (0gβπ
)(leβπ
)(π(.rβπ
)π)) |
73 | 54 | adantr 482 |
. . . . . . 7
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (0gβπ
) = (0gβ(π
βΎs π΄))) |
74 | 58 | adantr 482 |
. . . . . . 7
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (leβπ
) = (leβ(π
βΎs π΄))) |
75 | 55 | adantr 482 |
. . . . . . . . 9
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β π΄ β V) |
76 | 27, 70 | ressmulr 17058 |
. . . . . . . . 9
β’ (π΄ β V β
(.rβπ
) =
(.rβ(π
βΎs π΄))) |
77 | 75, 76 | syl 17 |
. . . . . . . 8
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (.rβπ
) = (.rβ(π
βΎs π΄))) |
78 | 77 | oveqd 7320 |
. . . . . . 7
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (π(.rβπ
)π) = (π(.rβ(π
βΎs π΄))π)) |
79 | 73, 74, 78 | breq123d 5095 |
. . . . . 6
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β ((0gβπ
)(leβπ
)(π(.rβπ
)π) β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))(π(.rβ(π
βΎs π΄))π))) |
80 | 72, 79 | mpbid 232 |
. . . . 5
β’
(((((π
β oRing
β§ (π
βΎs π΄)
β Ring) β§ π β
(Baseβ(π
βΎs π΄)))
β§ π β
(Baseβ(π
βΎs π΄)))
β§ ((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π)) β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))(π(.rβ(π
βΎs π΄))π)) |
81 | 80 | ex 414 |
. . . 4
β’ ((((π
β oRing β§ (π
βΎs π΄) β Ring) β§ π β (Baseβ(π
βΎs π΄))) β§ π β (Baseβ(π
βΎs π΄))) β (((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π) β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))(π(.rβ(π
βΎs π΄))π))) |
82 | 81 | anasss 468 |
. . 3
β’ (((π
β oRing β§ (π
βΎs π΄) β Ring) β§ (π β (Baseβ(π
βΎs π΄)) β§ π β (Baseβ(π
βΎs π΄)))) β (((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π) β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))(π(.rβ(π
βΎs π΄))π))) |
83 | 82 | ralrimivva 3194 |
. 2
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β
βπ β
(Baseβ(π
βΎs π΄))βπ β (Baseβ(π
βΎs π΄))(((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π) β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))(π(.rβ(π
βΎs π΄))π))) |
84 | | eqid 2736 |
. . 3
β’
(0gβ(π
βΎs π΄)) = (0gβ(π
βΎs π΄)) |
85 | | eqid 2736 |
. . 3
β’
(.rβ(π
βΎs π΄)) = (.rβ(π
βΎs π΄)) |
86 | | eqid 2736 |
. . 3
β’
(leβ(π
βΎs π΄)) =
(leβ(π
βΎs π΄)) |
87 | 20, 84, 85, 86 | isorng 31539 |
. 2
β’ ((π
βΎs π΄) β oRing β ((π
βΎs π΄) β Ring β§ (π
βΎs π΄) β oGrp β§
βπ β
(Baseβ(π
βΎs π΄))βπ β (Baseβ(π
βΎs π΄))(((0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π β§ (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))π) β (0gβ(π
βΎs π΄))(leβ(π
βΎs π΄))(π(.rβ(π
βΎs π΄))π)))) |
88 | 1, 12, 83, 87 | syl3anbrc 1343 |
1
β’ ((π
β oRing β§ (π
βΎs π΄) β Ring) β (π
βΎs π΄) β oRing) |