Step |
Hyp |
Ref |
Expression |
1 |
|
srgpcomp.s |
|- S = ( Base ` R ) |
2 |
|
srgpcomp.m |
|- .X. = ( .r ` R ) |
3 |
|
srgpcomp.g |
|- G = ( mulGrp ` R ) |
4 |
|
srgpcomp.e |
|- .^ = ( .g ` G ) |
5 |
|
srgpcomp.r |
|- ( ph -> R e. SRing ) |
6 |
|
srgpcomp.a |
|- ( ph -> A e. S ) |
7 |
|
srgpcomp.b |
|- ( ph -> B e. S ) |
8 |
|
srgpcomp.k |
|- ( ph -> K e. NN0 ) |
9 |
|
srgpcomp.c |
|- ( ph -> ( A .X. B ) = ( B .X. A ) ) |
10 |
|
srgpcompp.n |
|- ( ph -> N e. NN0 ) |
11 |
3 1
|
mgpbas |
|- S = ( Base ` G ) |
12 |
3
|
srgmgp |
|- ( R e. SRing -> G e. Mnd ) |
13 |
5 12
|
syl |
|- ( ph -> G e. Mnd ) |
14 |
11 4 13 10 6
|
mulgnn0cld |
|- ( ph -> ( N .^ A ) e. S ) |
15 |
11 4 13 8 7
|
mulgnn0cld |
|- ( ph -> ( K .^ B ) e. S ) |
16 |
1 2
|
srgass |
|- ( ( R e. SRing /\ ( ( N .^ A ) e. S /\ ( K .^ B ) e. S /\ A e. S ) ) -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) ) |
17 |
5 14 15 6 16
|
syl13anc |
|- ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) ) |
18 |
1 2 3 4 5 6 7 8 9
|
srgpcomp |
|- ( ph -> ( ( K .^ B ) .X. A ) = ( A .X. ( K .^ B ) ) ) |
19 |
18
|
oveq2d |
|- ( ph -> ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) = ( ( N .^ A ) .X. ( A .X. ( K .^ B ) ) ) ) |
20 |
1 2
|
srgass |
|- ( ( R e. SRing /\ ( ( N .^ A ) e. S /\ A e. S /\ ( K .^ B ) e. S ) ) -> ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) = ( ( N .^ A ) .X. ( A .X. ( K .^ B ) ) ) ) |
21 |
5 14 6 15 20
|
syl13anc |
|- ( ph -> ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) = ( ( N .^ A ) .X. ( A .X. ( K .^ B ) ) ) ) |
22 |
19 21
|
eqtr4d |
|- ( ph -> ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) = ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) ) |
23 |
3 2
|
mgpplusg |
|- .X. = ( +g ` G ) |
24 |
11 4 23
|
mulgnn0p1 |
|- ( ( G e. Mnd /\ N e. NN0 /\ A e. S ) -> ( ( N + 1 ) .^ A ) = ( ( N .^ A ) .X. A ) ) |
25 |
13 10 6 24
|
syl3anc |
|- ( ph -> ( ( N + 1 ) .^ A ) = ( ( N .^ A ) .X. A ) ) |
26 |
25
|
eqcomd |
|- ( ph -> ( ( N .^ A ) .X. A ) = ( ( N + 1 ) .^ A ) ) |
27 |
26
|
oveq1d |
|- ( ph -> ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) ) |
28 |
17 22 27
|
3eqtrd |
|- ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) ) |