Step | Hyp | Ref
| Expression |
1 | | ringneg.3 |
. . . . . . . . 9
β’ π = ran πΊ |
2 | | ringneg.1 |
. . . . . . . . . 10
β’ πΊ = (1st βπ
) |
3 | 2 | rneqi 5893 |
. . . . . . . . 9
β’ ran πΊ = ran (1st
βπ
) |
4 | 1, 3 | eqtri 2761 |
. . . . . . . 8
β’ π = ran (1st
βπ
) |
5 | | ringneg.2 |
. . . . . . . 8
β’ π» = (2nd βπ
) |
6 | | ringneg.5 |
. . . . . . . 8
β’ π = (GIdβπ») |
7 | 4, 5, 6 | rngo1cl 36444 |
. . . . . . 7
β’ (π
β RingOps β π β π) |
8 | | ringneg.4 |
. . . . . . . 8
β’ π = (invβπΊ) |
9 | 2, 1, 8 | rngonegcl 36432 |
. . . . . . 7
β’ ((π
β RingOps β§ π β π) β (πβπ) β π) |
10 | 7, 9 | mpdan 686 |
. . . . . 6
β’ (π
β RingOps β (πβπ) β π) |
11 | 10 | adantr 482 |
. . . . 5
β’ ((π
β RingOps β§ π΄ β π) β (πβπ) β π) |
12 | 7 | adantr 482 |
. . . . 5
β’ ((π
β RingOps β§ π΄ β π) β π β π) |
13 | 11, 12 | jca 513 |
. . . 4
β’ ((π
β RingOps β§ π΄ β π) β ((πβπ) β π β§ π β π)) |
14 | 2, 5, 1 | rngodi 36409 |
. . . . . 6
β’ ((π
β RingOps β§ (π΄ β π β§ (πβπ) β π β§ π β π)) β (π΄π»((πβπ)πΊπ)) = ((π΄π»(πβπ))πΊ(π΄π»π))) |
15 | 14 | 3exp2 1355 |
. . . . 5
β’ (π
β RingOps β (π΄ β π β ((πβπ) β π β (π β π β (π΄π»((πβπ)πΊπ)) = ((π΄π»(πβπ))πΊ(π΄π»π)))))) |
16 | 15 | imp43 429 |
. . . 4
β’ (((π
β RingOps β§ π΄ β π) β§ ((πβπ) β π β§ π β π)) β (π΄π»((πβπ)πΊπ)) = ((π΄π»(πβπ))πΊ(π΄π»π))) |
17 | 13, 16 | mpdan 686 |
. . 3
β’ ((π
β RingOps β§ π΄ β π) β (π΄π»((πβπ)πΊπ)) = ((π΄π»(πβπ))πΊ(π΄π»π))) |
18 | | eqid 2733 |
. . . . . . . 8
β’
(GIdβπΊ) =
(GIdβπΊ) |
19 | 2, 1, 8, 18 | rngoaddneg2 36434 |
. . . . . . 7
β’ ((π
β RingOps β§ π β π) β ((πβπ)πΊπ) = (GIdβπΊ)) |
20 | 7, 19 | mpdan 686 |
. . . . . 6
β’ (π
β RingOps β ((πβπ)πΊπ) = (GIdβπΊ)) |
21 | 20 | adantr 482 |
. . . . 5
β’ ((π
β RingOps β§ π΄ β π) β ((πβπ)πΊπ) = (GIdβπΊ)) |
22 | 21 | oveq2d 7374 |
. . . 4
β’ ((π
β RingOps β§ π΄ β π) β (π΄π»((πβπ)πΊπ)) = (π΄π»(GIdβπΊ))) |
23 | 18, 1, 2, 5 | rngorz 36428 |
. . . 4
β’ ((π
β RingOps β§ π΄ β π) β (π΄π»(GIdβπΊ)) = (GIdβπΊ)) |
24 | 22, 23 | eqtrd 2773 |
. . 3
β’ ((π
β RingOps β§ π΄ β π) β (π΄π»((πβπ)πΊπ)) = (GIdβπΊ)) |
25 | 5, 4, 6 | rngoridm 36443 |
. . . 4
β’ ((π
β RingOps β§ π΄ β π) β (π΄π»π) = π΄) |
26 | 25 | oveq2d 7374 |
. . 3
β’ ((π
β RingOps β§ π΄ β π) β ((π΄π»(πβπ))πΊ(π΄π»π)) = ((π΄π»(πβπ))πΊπ΄)) |
27 | 17, 24, 26 | 3eqtr3rd 2782 |
. 2
β’ ((π
β RingOps β§ π΄ β π) β ((π΄π»(πβπ))πΊπ΄) = (GIdβπΊ)) |
28 | 2, 5, 1 | rngocl 36406 |
. . . 4
β’ ((π
β RingOps β§ π΄ β π β§ (πβπ) β π) β (π΄π»(πβπ)) β π) |
29 | 11, 28 | mpd3an3 1463 |
. . 3
β’ ((π
β RingOps β§ π΄ β π) β (π΄π»(πβπ)) β π) |
30 | 2 | rngogrpo 36415 |
. . . 4
β’ (π
β RingOps β πΊ β GrpOp) |
31 | 1, 18, 8 | grpoinvid2 29513 |
. . . 4
β’ ((πΊ β GrpOp β§ π΄ β π β§ (π΄π»(πβπ)) β π) β ((πβπ΄) = (π΄π»(πβπ)) β ((π΄π»(πβπ))πΊπ΄) = (GIdβπΊ))) |
32 | 30, 31 | syl3an1 1164 |
. . 3
β’ ((π
β RingOps β§ π΄ β π β§ (π΄π»(πβπ)) β π) β ((πβπ΄) = (π΄π»(πβπ)) β ((π΄π»(πβπ))πΊπ΄) = (GIdβπΊ))) |
33 | 29, 32 | mpd3an3 1463 |
. 2
β’ ((π
β RingOps β§ π΄ β π) β ((πβπ΄) = (π΄π»(πβπ)) β ((π΄π»(πβπ))πΊπ΄) = (GIdβπΊ))) |
34 | 27, 33 | mpbird 257 |
1
β’ ((π
β RingOps β§ π΄ β π) β (πβπ΄) = (π΄π»(πβπ))) |