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 × ˙ = R
srgpcomp.g G = mulGrp R
srgpcomp.e × ˙ = G
srgpcomp.r φ R SRing
srgpcomp.a φ A S
srgpcomp.b φ B S
srgpcomp.k φ K 0
srgpcomp.c φ A × ˙ B = B × ˙ A
srgpcompp.n φ N 0
Assertion srgpcompp φ N × ˙ A × ˙ K × ˙ B × ˙ A = N + 1 × ˙ A × ˙ K × ˙ B

Proof

Step Hyp Ref Expression
1 srgpcomp.s S = Base R
2 srgpcomp.m × ˙ = R
3 srgpcomp.g G = mulGrp R
4 srgpcomp.e × ˙ = G
5 srgpcomp.r φ R SRing
6 srgpcomp.a φ A S
7 srgpcomp.b φ B S
8 srgpcomp.k φ K 0
9 srgpcomp.c φ A × ˙ B = B × ˙ A
10 srgpcompp.n φ N 0
11 3 1 mgpbas S = Base G
12 3 srgmgp R SRing G Mnd
13 5 12 syl φ G Mnd
14 11 4 13 10 6 mulgnn0cld φ N × ˙ A S
15 11 4 13 8 7 mulgnn0cld φ K × ˙ B S
16 1 2 srgass R SRing N × ˙ A S K × ˙ B S A S N × ˙ A × ˙ K × ˙ B × ˙ A = N × ˙ A × ˙ K × ˙ B × ˙ A
17 5 14 15 6 16 syl13anc φ N × ˙ A × ˙ K × ˙ B × ˙ A = N × ˙ A × ˙ K × ˙ B × ˙ A
18 1 2 3 4 5 6 7 8 9 srgpcomp φ K × ˙ B × ˙ A = A × ˙ K × ˙ B
19 18 oveq2d φ N × ˙ A × ˙ K × ˙ B × ˙ A = N × ˙ A × ˙ A × ˙ K × ˙ B
20 1 2 srgass R SRing N × ˙ A S A S K × ˙ B S N × ˙ A × ˙ A × ˙ K × ˙ B = N × ˙ A × ˙ A × ˙ K × ˙ B
21 5 14 6 15 20 syl13anc φ N × ˙ A × ˙ A × ˙ K × ˙ B = N × ˙ A × ˙ A × ˙ K × ˙ B
22 19 21 eqtr4d φ N × ˙ A × ˙ K × ˙ B × ˙ A = N × ˙ A × ˙ A × ˙ K × ˙ B
23 3 2 mgpplusg × ˙ = + G
24 11 4 23 mulgnn0p1 G Mnd N 0 A S N + 1 × ˙ A = N × ˙ A × ˙ A
25 13 10 6 24 syl3anc φ N + 1 × ˙ A = N × ˙ A × ˙ A
26 25 eqcomd φ N × ˙ A × ˙ A = N + 1 × ˙ A
27 26 oveq1d φ N × ˙ A × ˙ A × ˙ K × ˙ B = N + 1 × ˙ A × ˙ K × ˙ B
28 17 22 27 3eqtrd φ N × ˙ A × ˙ K × ˙ B × ˙ A = N + 1 × ˙ A × ˙ K × ˙ B