Step |
Hyp |
Ref |
Expression |
1 |
|
prdscmnd.y |
|- Y = ( S Xs_ R ) |
2 |
|
prdscmnd.i |
|- ( ph -> I e. W ) |
3 |
|
prdscmnd.s |
|- ( ph -> S e. V ) |
4 |
|
prdsgabld.r |
|- ( ph -> R : I --> Abel ) |
5 |
|
ablgrp |
|- ( a e. Abel -> a e. Grp ) |
6 |
5
|
ssriv |
|- Abel C_ Grp |
7 |
|
fss |
|- ( ( R : I --> Abel /\ Abel C_ Grp ) -> R : I --> Grp ) |
8 |
4 6 7
|
sylancl |
|- ( ph -> R : I --> Grp ) |
9 |
1 2 3 8
|
prdsgrpd |
|- ( ph -> Y e. Grp ) |
10 |
|
ablcmn |
|- ( a e. Abel -> a e. CMnd ) |
11 |
10
|
ssriv |
|- Abel C_ CMnd |
12 |
|
fss |
|- ( ( R : I --> Abel /\ Abel C_ CMnd ) -> R : I --> CMnd ) |
13 |
4 11 12
|
sylancl |
|- ( ph -> R : I --> CMnd ) |
14 |
1 2 3 13
|
prdscmnd |
|- ( ph -> Y e. CMnd ) |
15 |
|
isabl |
|- ( Y e. Abel <-> ( Y e. Grp /\ Y e. CMnd ) ) |
16 |
9 14 15
|
sylanbrc |
|- ( ph -> Y e. Abel ) |