Metamath Proof Explorer

Theorem srgpcomppsc

Description: If two elements of a semiring commute, they also commute if the elements are raised to a higher power and a scalar multiplication is involved. (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 )
srgpcomppsc.t 
|- .x. = ( .g ` R )
srgpcomppsc.c 
|- ( ph -> C e. NN0 )
Assertion srgpcomppsc 
|- ( ph -> ( ( C .x. ( ( N .^ A ) .X. ( K .^ B ) ) ) .X. A ) = ( C .x. ( ( ( 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 srgpcomppsc.t 
 |-  .x. = ( .g ` R )
12 srgpcomppsc.c 
 |-  ( ph -> C e. NN0 )
13 3 srgmgp 
 |-  ( R e. SRing -> G e. Mnd )
14 5 13 syl 
 |-  ( ph -> G e. Mnd )
15 3 1 mgpbas 
 |-  S = ( Base ` G )
16 15 4 mulgnn0cl 
 |-  ( ( G e. Mnd /\ N e. NN0 /\ A e. S ) -> ( N .^ A ) e. S )
17 14 10 6 16 syl3anc 
 |-  ( ph -> ( N .^ A ) e. S )
18 15 4 mulgnn0cl 
 |-  ( ( G e. Mnd /\ K e. NN0 /\ B e. S ) -> ( K .^ B ) e. S )
19 14 8 7 18 syl3anc 
 |-  ( ph -> ( K .^ B ) e. S )
20 1 11 2 srgmulgass 
 |-  ( ( R e. SRing /\ ( C e. NN0 /\ ( N .^ A ) e. S /\ ( K .^ B ) e. S ) ) -> ( ( C .x. ( N .^ A ) ) .X. ( K .^ B ) ) = ( C .x. ( ( N .^ A ) .X. ( K .^ B ) ) ) )
21 20 eqcomd 
 |-  ( ( R e. SRing /\ ( C e. NN0 /\ ( N .^ A ) e. S /\ ( K .^ B ) e. S ) ) -> ( C .x. ( ( N .^ A ) .X. ( K .^ B ) ) ) = ( ( C .x. ( N .^ A ) ) .X. ( K .^ B ) ) )
22 5 12 17 19 21 syl13anc 
 |-  ( ph -> ( C .x. ( ( N .^ A ) .X. ( K .^ B ) ) ) = ( ( C .x. ( N .^ A ) ) .X. ( K .^ B ) ) )
23 22 oveq1d 
 |-  ( ph -> ( ( C .x. ( ( N .^ A ) .X. ( K .^ B ) ) ) .X. A ) = ( ( ( C .x. ( N .^ A ) ) .X. ( K .^ B ) ) .X. A ) )
24 srgmnd 
 |-  ( R e. SRing -> R e. Mnd )
25 5 24 syl 
 |-  ( ph -> R e. Mnd )
26 1 11 mulgnn0cl 
 |-  ( ( R e. Mnd /\ C e. NN0 /\ ( N .^ A ) e. S ) -> ( C .x. ( N .^ A ) ) e. S )
27 25 12 17 26 syl3anc 
 |-  ( ph -> ( C .x. ( N .^ A ) ) e. S )
28 1 2 srgass 
 |-  ( ( R e. SRing /\ ( ( C .x. ( N .^ A ) ) e. S /\ ( K .^ B ) e. S /\ A e. S ) ) -> ( ( ( C .x. ( N .^ A ) ) .X. ( K .^ B ) ) .X. A ) = ( ( C .x. ( N .^ A ) ) .X. ( ( K .^ B ) .X. A ) ) )
29 5 27 19 6 28 syl13anc 
 |-  ( ph -> ( ( ( C .x. ( N .^ A ) ) .X. ( K .^ B ) ) .X. A ) = ( ( C .x. ( N .^ A ) ) .X. ( ( K .^ B ) .X. A ) ) )
30 23 29 eqtrd 
 |-  ( ph -> ( ( C .x. ( ( N .^ A ) .X. ( K .^ B ) ) ) .X. A ) = ( ( C .x. ( N .^ A ) ) .X. ( ( K .^ B ) .X. A ) ) )
31 1 2 srgcl 
 |-  ( ( R e. SRing /\ ( K .^ B ) e. S /\ A e. S ) -> ( ( K .^ B ) .X. A ) e. S )
32 5 19 6 31 syl3anc 
 |-  ( ph -> ( ( K .^ B ) .X. A ) e. S )
33 1 11 2 srgmulgass 
 |-  ( ( R e. SRing /\ ( C e. NN0 /\ ( N .^ A ) e. S /\ ( ( K .^ B ) .X. A ) e. S ) ) -> ( ( C .x. ( N .^ A ) ) .X. ( ( K .^ B ) .X. A ) ) = ( C .x. ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) ) )
34 5 12 17 32 33 syl13anc 
 |-  ( ph -> ( ( C .x. ( N .^ A ) ) .X. ( ( K .^ B ) .X. A ) ) = ( C .x. ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) ) )
35 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 ) ) )
36 5 17 19 6 35 syl13anc 
 |-  ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) )
37 36 eqcomd 
 |-  ( ph -> ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) = ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) )
38 37 oveq2d 
 |-  ( ph -> ( C .x. ( ( N .^ A ) .X. ( ( K .^ B ) .X. A ) ) ) = ( C .x. ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) ) )
39 34 38 eqtrd 
 |-  ( ph -> ( ( C .x. ( N .^ A ) ) .X. ( ( K .^ B ) .X. A ) ) = ( C .x. ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) ) )
40 1 2 3 4 5 6 7 8 9 10 srgpcompp 
 |-  ( ph -> ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) = ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) )
41 40 oveq2d 
 |-  ( ph -> ( C .x. ( ( ( N .^ A ) .X. ( K .^ B ) ) .X. A ) ) = ( C .x. ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) ) )
42 30 39 41 3eqtrd 
 |-  ( ph -> ( ( C .x. ( ( N .^ A ) .X. ( K .^ B ) ) ) .X. A ) = ( C .x. ( ( ( N + 1 ) .^ A ) .X. ( K .^ B ) ) ) )