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 ‘ 𝑅 )
srgpcomp.g ⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
srgpcomp.e ⊢ ↑ = ( .g ‘ 𝐺 )
srgpcomp.r ⊢ ( 𝜑 → 𝑅 ∈ SRing )
srgpcomp.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
srgpcomp.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
srgpcomp.k ⊢ ( 𝜑 → 𝐾 ∈ ℕ0 )
srgpcomp.c ⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) = ( 𝐵 × 𝐴 ) )
srgpcompp.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
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 ‘ 𝑅 )
3 srgpcomp.g ⊢ 𝐺 = ( mulGrp ‘ 𝑅 )
4 srgpcomp.e ⊢ ↑ = ( .g ‘ 𝐺 )
5 srgpcomp.r ⊢ ( 𝜑 → 𝑅 ∈ SRing )
6 srgpcomp.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
7 srgpcomp.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
8 srgpcomp.k ⊢ ( 𝜑 → 𝐾 ∈ ℕ0 )
9 srgpcomp.c ⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) = ( 𝐵 × 𝐴 ) )
10 srgpcompp.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
11 3 srgmgp ⊢ ( 𝑅 ∈ SRing → 𝐺 ∈ Mnd )
12 5 11 syl ⊢ ( 𝜑 → 𝐺 ∈ Mnd )
13 3 1 mgpbas ⊢ 𝑆 = ( Base ‘ 𝐺 )
14 13 4 mulgnn0cl ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆 ) → ( 𝑁 ↑ 𝐴 ) ∈ 𝑆 )
15 12 10 6 14 syl3anc ⊢ ( 𝜑 → ( 𝑁 ↑ 𝐴 ) ∈ 𝑆 )
16 13 4 mulgnn0cl ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐾 ∈ ℕ0 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐾 ↑ 𝐵 ) ∈ 𝑆 )
17 12 8 7 16 syl3anc ⊢ ( 𝜑 → ( 𝐾 ↑ 𝐵 ) ∈ 𝑆 )
18 1 2 srgass ⊢ ( ( 𝑅 ∈ SRing ∧ ( ( 𝑁 ↑ 𝐴 ) ∈ 𝑆 ∧ ( 𝐾 ↑ 𝐵 ) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) ) → ( ( ( 𝑁 ↑ 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) × 𝐴 ) = ( ( 𝑁 ↑ 𝐴 ) × ( ( 𝐾 ↑ 𝐵 ) × 𝐴 ) ) )
19 5 15 17 6 18 syl13anc ⊢ ( 𝜑 → ( ( ( 𝑁 ↑ 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) × 𝐴 ) = ( ( 𝑁 ↑ 𝐴 ) × ( ( 𝐾 ↑ 𝐵 ) × 𝐴 ) ) )
20 1 2 3 4 5 6 7 8 9 srgpcomp ⊢ ( 𝜑 → ( ( 𝐾 ↑ 𝐵 ) × 𝐴 ) = ( 𝐴 × ( 𝐾 ↑ 𝐵 ) ) )
21 20 oveq2d ⊢ ( 𝜑 → ( ( 𝑁 ↑ 𝐴 ) × ( ( 𝐾 ↑ 𝐵 ) × 𝐴 ) ) = ( ( 𝑁 ↑ 𝐴 ) × ( 𝐴 × ( 𝐾 ↑ 𝐵 ) ) ) )
22 1 2 srgass ⊢ ( ( 𝑅 ∈ SRing ∧ ( ( 𝑁 ↑ 𝐴 ) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ ( 𝐾 ↑ 𝐵 ) ∈ 𝑆 ) ) → ( ( ( 𝑁 ↑ 𝐴 ) × 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) = ( ( 𝑁 ↑ 𝐴 ) × ( 𝐴 × ( 𝐾 ↑ 𝐵 ) ) ) )
23 5 15 6 17 22 syl13anc ⊢ ( 𝜑 → ( ( ( 𝑁 ↑ 𝐴 ) × 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) = ( ( 𝑁 ↑ 𝐴 ) × ( 𝐴 × ( 𝐾 ↑ 𝐵 ) ) ) )
24 21 23 eqtr4d ⊢ ( 𝜑 → ( ( 𝑁 ↑ 𝐴 ) × ( ( 𝐾 ↑ 𝐵 ) × 𝐴 ) ) = ( ( ( 𝑁 ↑ 𝐴 ) × 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) )
25 3 2 mgpplusg ⊢ × = ( +g ‘ 𝐺 )
26 13 4 25 mulgnn0p1 ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆 ) → ( ( 𝑁 + 1 ) ↑ 𝐴 ) = ( ( 𝑁 ↑ 𝐴 ) × 𝐴 ) )
27 12 10 6 26 syl3anc ⊢ ( 𝜑 → ( ( 𝑁 + 1 ) ↑ 𝐴 ) = ( ( 𝑁 ↑ 𝐴 ) × 𝐴 ) )
28 27 eqcomd ⊢ ( 𝜑 → ( ( 𝑁 ↑ 𝐴 ) × 𝐴 ) = ( ( 𝑁 + 1 ) ↑ 𝐴 ) )
29 28 oveq1d ⊢ ( 𝜑 → ( ( ( 𝑁 ↑ 𝐴 ) × 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) = ( ( ( 𝑁 + 1 ) ↑ 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) )
30 19 24 29 3eqtrd ⊢ ( 𝜑 → ( ( ( 𝑁 ↑ 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) × 𝐴 ) = ( ( ( 𝑁 + 1 ) ↑ 𝐴 ) × ( 𝐾 ↑ 𝐵 ) ) )