Step | Hyp | Ref
| Expression |
1 | | ringnegmul.3 |
. . . . . . 7
β’ π = ran πΊ |
2 | | ringnegmul.1 |
. . . . . . . 8
β’ πΊ = (1st βπ
) |
3 | 2 | rneqi 5937 |
. . . . . . 7
β’ ran πΊ = ran (1st
βπ
) |
4 | 1, 3 | eqtri 2761 |
. . . . . 6
β’ π = ran (1st
βπ
) |
5 | | ringnegmul.2 |
. . . . . 6
β’ π» = (2nd βπ
) |
6 | | eqid 2733 |
. . . . . 6
β’
(GIdβπ») =
(GIdβπ») |
7 | 4, 5, 6 | rngo1cl 36807 |
. . . . 5
β’ (π
β RingOps β
(GIdβπ») β π) |
8 | | ringnegmul.4 |
. . . . . 6
β’ π = (invβπΊ) |
9 | 2, 1, 8 | rngonegcl 36795 |
. . . . 5
β’ ((π
β RingOps β§
(GIdβπ») β π) β (πβ(GIdβπ»)) β π) |
10 | 7, 9 | mpdan 686 |
. . . 4
β’ (π
β RingOps β (πβ(GIdβπ»)) β π) |
11 | 2, 5, 1 | rngoass 36774 |
. . . . 5
β’ ((π
β RingOps β§ ((πβ(GIdβπ»)) β π β§ π΄ β π β§ π΅ β π)) β (((πβ(GIdβπ»))π»π΄)π»π΅) = ((πβ(GIdβπ»))π»(π΄π»π΅))) |
12 | 11 | 3exp2 1355 |
. . . 4
β’ (π
β RingOps β ((πβ(GIdβπ»)) β π β (π΄ β π β (π΅ β π β (((πβ(GIdβπ»))π»π΄)π»π΅) = ((πβ(GIdβπ»))π»(π΄π»π΅)))))) |
13 | 10, 12 | mpd 15 |
. . 3
β’ (π
β RingOps β (π΄ β π β (π΅ β π β (((πβ(GIdβπ»))π»π΄)π»π΅) = ((πβ(GIdβπ»))π»(π΄π»π΅))))) |
14 | 13 | 3imp 1112 |
. 2
β’ ((π
β RingOps β§ π΄ β π β§ π΅ β π) β (((πβ(GIdβπ»))π»π΄)π»π΅) = ((πβ(GIdβπ»))π»(π΄π»π΅))) |
15 | 2, 5, 1, 8, 6 | rngonegmn1l 36809 |
. . . 4
β’ ((π
β RingOps β§ π΄ β π) β (πβπ΄) = ((πβ(GIdβπ»))π»π΄)) |
16 | 15 | 3adant3 1133 |
. . 3
β’ ((π
β RingOps β§ π΄ β π β§ π΅ β π) β (πβπ΄) = ((πβ(GIdβπ»))π»π΄)) |
17 | 16 | oveq1d 7424 |
. 2
β’ ((π
β RingOps β§ π΄ β π β§ π΅ β π) β ((πβπ΄)π»π΅) = (((πβ(GIdβπ»))π»π΄)π»π΅)) |
18 | 2, 5, 1 | rngocl 36769 |
. . . . 5
β’ ((π
β RingOps β§ π΄ β π β§ π΅ β π) β (π΄π»π΅) β π) |
19 | 18 | 3expb 1121 |
. . . 4
β’ ((π
β RingOps β§ (π΄ β π β§ π΅ β π)) β (π΄π»π΅) β π) |
20 | 2, 5, 1, 8, 6 | rngonegmn1l 36809 |
. . . 4
β’ ((π
β RingOps β§ (π΄π»π΅) β π) β (πβ(π΄π»π΅)) = ((πβ(GIdβπ»))π»(π΄π»π΅))) |
21 | 19, 20 | syldan 592 |
. . 3
β’ ((π
β RingOps β§ (π΄ β π β§ π΅ β π)) β (πβ(π΄π»π΅)) = ((πβ(GIdβπ»))π»(π΄π»π΅))) |
22 | 21 | 3impb 1116 |
. 2
β’ ((π
β RingOps β§ π΄ β π β§ π΅ β π) β (πβ(π΄π»π΅)) = ((πβ(GIdβπ»))π»(π΄π»π΅))) |
23 | 14, 17, 22 | 3eqtr4rd 2784 |
1
β’ ((π
β RingOps β§ π΄ β π β§ π΅ β π) β (πβ(π΄π»π΅)) = ((πβπ΄)π»π΅)) |