| Step |
Hyp |
Ref |
Expression |
| 1 |
|
subgmulgcld.b |
|- B = ( Base ` R ) |
| 2 |
|
subgmulgcld.x |
|- .x. = ( .g ` R ) |
| 3 |
|
subgmulgcld.r |
|- ( ph -> R e. Grp ) |
| 4 |
|
subgmulgcld.a |
|- ( ph -> A e. S ) |
| 5 |
|
subgmulgcld.s |
|- ( ph -> S e. ( SubGrp ` R ) ) |
| 6 |
|
subgmulgcld.z |
|- ( ph -> Z e. ZZ ) |
| 7 |
|
eqid |
|- ( Base ` ( R |`s S ) ) = ( Base ` ( R |`s S ) ) |
| 8 |
|
eqid |
|- ( .g ` ( R |`s S ) ) = ( .g ` ( R |`s S ) ) |
| 9 |
|
eqid |
|- ( R |`s S ) = ( R |`s S ) |
| 10 |
9
|
subggrp |
|- ( S e. ( SubGrp ` R ) -> ( R |`s S ) e. Grp ) |
| 11 |
5 10
|
syl |
|- ( ph -> ( R |`s S ) e. Grp ) |
| 12 |
1
|
subgss |
|- ( S e. ( SubGrp ` R ) -> S C_ B ) |
| 13 |
9 1
|
ressbas2 |
|- ( S C_ B -> S = ( Base ` ( R |`s S ) ) ) |
| 14 |
5 12 13
|
3syl |
|- ( ph -> S = ( Base ` ( R |`s S ) ) ) |
| 15 |
4 14
|
eleqtrd |
|- ( ph -> A e. ( Base ` ( R |`s S ) ) ) |
| 16 |
7 8 11 6 15
|
mulgcld |
|- ( ph -> ( Z ( .g ` ( R |`s S ) ) A ) e. ( Base ` ( R |`s S ) ) ) |
| 17 |
2 9 8
|
subgmulg |
|- ( ( S e. ( SubGrp ` R ) /\ Z e. ZZ /\ A e. S ) -> ( Z .x. A ) = ( Z ( .g ` ( R |`s S ) ) A ) ) |
| 18 |
5 6 4 17
|
syl3anc |
|- ( ph -> ( Z .x. A ) = ( Z ( .g ` ( R |`s S ) ) A ) ) |
| 19 |
16 18 14
|
3eltr4d |
|- ( ph -> ( Z .x. A ) e. S ) |