Step |
Hyp |
Ref |
Expression |
1 |
|
gsumccat.b |
|- B = ( Base ` G ) |
2 |
|
gsumccat.p |
|- .+ = ( +g ` G ) |
3 |
|
s1cl |
|- ( Z e. B -> <" Z "> e. Word B ) |
4 |
1 2
|
gsumccat |
|- ( ( G e. Mnd /\ W e. Word B /\ <" Z "> e. Word B ) -> ( G gsum ( W ++ <" Z "> ) ) = ( ( G gsum W ) .+ ( G gsum <" Z "> ) ) ) |
5 |
3 4
|
syl3an3 |
|- ( ( G e. Mnd /\ W e. Word B /\ Z e. B ) -> ( G gsum ( W ++ <" Z "> ) ) = ( ( G gsum W ) .+ ( G gsum <" Z "> ) ) ) |
6 |
1
|
gsumws1 |
|- ( Z e. B -> ( G gsum <" Z "> ) = Z ) |
7 |
6
|
3ad2ant3 |
|- ( ( G e. Mnd /\ W e. Word B /\ Z e. B ) -> ( G gsum <" Z "> ) = Z ) |
8 |
7
|
oveq2d |
|- ( ( G e. Mnd /\ W e. Word B /\ Z e. B ) -> ( ( G gsum W ) .+ ( G gsum <" Z "> ) ) = ( ( G gsum W ) .+ Z ) ) |
9 |
5 8
|
eqtrd |
|- ( ( G e. Mnd /\ W e. Word B /\ Z e. B ) -> ( G gsum ( W ++ <" Z "> ) ) = ( ( G gsum W ) .+ Z ) ) |