Metamath Proof Explorer


Theorem cznabel

Description: The ring constructed from a Z/nZ structure by replacing the (multiplicative) ring operation by a constant operation is an abelian group. (Contributed by AV, 16-Feb-2020)

Ref Expression
Hypotheses cznrng.y 𝑌 = ( ℤ/nℤ ‘ 𝑁 )
cznrng.b 𝐵 = ( Base ‘ 𝑌 )
cznrng.x 𝑋 = ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵𝐶 ) ⟩ )
Assertion cznabel ( ( 𝑁 ∈ ℕ ∧ 𝐶𝐵 ) → 𝑋 ∈ Abel )

Proof

Step Hyp Ref Expression
1 cznrng.y 𝑌 = ( ℤ/nℤ ‘ 𝑁 )
2 cznrng.b 𝐵 = ( Base ‘ 𝑌 )
3 cznrng.x 𝑋 = ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵𝐶 ) ⟩ )
4 nnnn0 ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 )
5 4 adantr ( ( 𝑁 ∈ ℕ ∧ 𝐶𝐵 ) → 𝑁 ∈ ℕ0 )
6 1 zncrng ( 𝑁 ∈ ℕ0𝑌 ∈ CRing )
7 5 6 syl ( ( 𝑁 ∈ ℕ ∧ 𝐶𝐵 ) → 𝑌 ∈ CRing )
8 crngring ( 𝑌 ∈ CRing → 𝑌 ∈ Ring )
9 ringabl ( 𝑌 ∈ Ring → 𝑌 ∈ Abel )
10 7 8 9 3syl ( ( 𝑁 ∈ ℕ ∧ 𝐶𝐵 ) → 𝑌 ∈ Abel )
11 3 fveq2i ( Base ‘ 𝑋 ) = ( Base ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵𝐶 ) ⟩ ) )
12 baseid Base = Slot ( Base ‘ ndx )
13 basendxnmulrndx ( Base ‘ ndx ) ≠ ( .r ‘ ndx )
14 12 13 setsnid ( Base ‘ 𝑌 ) = ( Base ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵𝐶 ) ⟩ ) )
15 11 14 eqtr4i ( Base ‘ 𝑋 ) = ( Base ‘ 𝑌 )
16 3 fveq2i ( +g𝑋 ) = ( +g ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵𝐶 ) ⟩ ) )
17 plusgid +g = Slot ( +g ‘ ndx )
18 plusgndxnmulrndx ( +g ‘ ndx ) ≠ ( .r ‘ ndx )
19 17 18 setsnid ( +g𝑌 ) = ( +g ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵𝐶 ) ⟩ ) )
20 16 19 eqtr4i ( +g𝑋 ) = ( +g𝑌 )
21 15 20 ablprop ( 𝑋 ∈ Abel ↔ 𝑌 ∈ Abel )
22 10 21 sylibr ( ( 𝑁 ∈ ℕ ∧ 𝐶𝐵 ) → 𝑋 ∈ Abel )