Metamath Proof Explorer

Theorem srgpcompp

Description: If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019)

Ref Expression
Hypotheses srgpcomp.s 
|- S = ( Base ` R )
srgpcomp.m 
|- .X. = ( .r ` R )
srgpcomp.g 
|- G = ( mulGrp ` R )
srgpcomp.e 
|- .^ = ( .g ` G )
srgpcomp.r 
|- ( ph -> R e. SRing )
srgpcomp.a 
|- ( ph -> A e. S )
srgpcomp.b 
|- ( ph -> B e. S )
srgpcomp.k 
|- ( ph -> K e. NN0 )
srgpcomp.c 
|- ( ph -> ( A .X. B ) = ( B .X. A ) )
srgpcompp.n 
|- ( ph -> N e. NN0 )
Assertion srgpcompp 
|- ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) )

Proof

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 srgmgp 
 |-  ( R e. SRing -> G e. Mnd )
12 5 11 syl 
 |-  ( ph -> G e. Mnd )
13 3 1 mgpbas 
 |-  S = ( Base ` G )
14 13 4 mulgnn0cl 
 |-  ( ( G e. Mnd /\ N e. NN0 /\ A e. S ) -> ( N .^ A ) e. S )
15 12 10 6 14 syl3anc 
 |-  ( ph -> ( N .^ A ) e. S )
16 13 4 mulgnn0cl 
 |-  ( ( G e. Mnd /\ K e. NN0 /\ B e. S ) -> ( K .^ B ) e. S )
17 12 8 7 16 syl3anc 
 |-  ( ph -> ( K .^ B ) e. S )
18 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 ) ) )
19 5 15 17 6 18 syl13anc 
 |-  ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) )
20 1 2 3 4 5 6 7 8 9 srgpcomp 
 |-  ( ph -> ( ( K .^ B ) .X. A ) = ( A .X. ( K .^ B ) ) )
21 20 oveq2d 
 |-  ( ph -> ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) = ( ( N .^ A ) .X. ( A .X. ( K .^ B ) ) ) )
22 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 ) ) ) )
23 5 15 6 17 22 syl13anc 
 |-  ( ph -> ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) = ( ( N .^ A ) .X. ( A .X. ( K .^ B ) ) ) )
24 21 23 eqtr4d 
 |-  ( ph -> ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) = ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) )
25 3 2 mgpplusg 
 |-  .X. = ( +g ` G )
26 13 4 25 mulgnn0p1 
 |-  ( ( G e. Mnd /\ N e. NN0 /\ A e. S ) -> ( ( N + 1 ) .^ A ) = ( ( N .^ A ) .X. A ) )
27 12 10 6 26 syl3anc 
 |-  ( ph -> ( ( N + 1 ) .^ A ) = ( ( N .^ A ) .X. A ) )
28 27 eqcomd 
 |-  ( ph -> ( ( N .^ A ) .X. A ) = ( ( N + 1 ) .^ A ) )
29 28 oveq1d 
 |-  ( ph -> ( ( ( N .^ A ) .X. A ) .X. ( K .^ B ) ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) )
30 19 24 29 3eqtrd 
 |-  ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) )