Metamath Proof Explorer

Theorem cznnring

Description: The ring constructed from a Z/nZ structure with 1 < n by replacing the (multiplicative) ring operation by a constant operation is not a unital ring. (Contributed by AV, 17-Feb-2020)

Ref Expression
Hypotheses cznrng.y ⊢ 𝑌 = ( ℤ/nℤ ‘ 𝑁 )
cznrng.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
cznrng.x ⊢ 𝑋 = ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ )
cznrng.0 ⊢ 0 = ( 0g ‘ 𝑌 )
Assertion cznnring ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐶 ∈ 𝐵 ) → 𝑋 ∉ Ring )

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 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ )
4 cznrng.0 ⊢ 0 = ( 0g ‘ 𝑌 )
5 eqid ⊢ ( mulGrp ‘ 𝑋 ) = ( mulGrp ‘ 𝑋 )
6 1 2 3 cznrnglem ⊢ 𝐵 = ( Base ‘ 𝑋 )
7 5 6 mgpbas ⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑋 ) )
8 3 fveq2i ⊢ ( mulGrp ‘ 𝑋 ) = ( mulGrp ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) )
9 1 fvexi ⊢ 𝑌 ∈ V
10 2 fvexi ⊢ 𝐵 ∈ V
11 10 10 mpoex ⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ∈ V
12 mulrid ⊢ .r = Slot ( .r ‘ ndx )
13 12 setsid ⊢ ( ( 𝑌 ∈ V ∧ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ∈ V ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) )
14 9 11 13 mp2an ⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) )
15 8 14 mgpplusg ⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( +g ‘ ( mulGrp ‘ 𝑋 ) )
16 15 eqcomi ⊢ ( +g ‘ ( mulGrp ‘ 𝑋 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 )
17 simpr ⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐶 ∈ 𝐵 ) → 𝐶 ∈ 𝐵 )
18 eluz2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁 ) )
19 1lt2 ⊢ 1 < 2
20 1red ⊢ ( 𝑁 ∈ ℤ → 1 ∈ ℝ )
21 2re ⊢ 2 ∈ ℝ
22 21 a1i ⊢ ( 𝑁 ∈ ℤ → 2 ∈ ℝ )
23 zre ⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ )
24 ltletr ⊢ ( ( 1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( 1 < 2 ∧ 2 ≤ 𝑁 ) → 1 < 𝑁 ) )
25 20 22 23 24 syl3anc ⊢ ( 𝑁 ∈ ℤ → ( ( 1 < 2 ∧ 2 ≤ 𝑁 ) → 1 < 𝑁 ) )
26 25 expcomd ⊢ ( 𝑁 ∈ ℤ → ( 2 ≤ 𝑁 → ( 1 < 2 → 1 < 𝑁 ) ) )
27 26 a1i ⊢ ( 2 ∈ ℤ → ( 𝑁 ∈ ℤ → ( 2 ≤ 𝑁 → ( 1 < 2 → 1 < 𝑁 ) ) ) )
28 27 3imp ⊢ ( ( 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁 ) → ( 1 < 2 → 1 < 𝑁 ) )
29 19 28 mpi ⊢ ( ( 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁 ) → 1 < 𝑁 )
30 18 29 sylbi ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) → 1 < 𝑁 )
31 eluz2nn ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) → 𝑁 ∈ ℕ )
32 1 2 znhash ⊢ ( 𝑁 ∈ ℕ → ( ♯ ‘ 𝐵 ) = 𝑁 )
33 31 32 syl ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) → ( ♯ ‘ 𝐵 ) = 𝑁 )
34 30 33 breqtrrd ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) → 1 < ( ♯ ‘ 𝐵 ) )
35 34 adantr ⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐶 ∈ 𝐵 ) → 1 < ( ♯ ‘ 𝐵 ) )
36 7 16 17 35 copisnmnd ⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐶 ∈ 𝐵 ) → ( mulGrp ‘ 𝑋 ) ∉ Mnd )
37 df-nel ⊢ ( ( mulGrp ‘ 𝑋 ) ∉ Mnd ↔ ¬ ( mulGrp ‘ 𝑋 ) ∈ Mnd )
38 36 37 sylib ⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐶 ∈ 𝐵 ) → ¬ ( mulGrp ‘ 𝑋 ) ∈ Mnd )
39 38 intn3an2d ⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐶 ∈ 𝐵 ) → ¬ ( 𝑋 ∈ Grp ∧ ( mulGrp ‘ 𝑋 ) ∈ Mnd ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) ( 𝑏 ( +g ‘ 𝑋 ) 𝑐 ) ) = ( ( 𝑎 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑏 ) ( +g ‘ 𝑋 ) ( 𝑎 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑐 ) ) ∧ ( ( 𝑎 ( +g ‘ 𝑋 ) 𝑏 ) ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑐 ) = ( ( 𝑎 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑐 ) ( +g ‘ 𝑋 ) ( 𝑏 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑐 ) ) ) ) )
40 eqid ⊢ ( +g ‘ 𝑋 ) = ( +g ‘ 𝑋 )
41 3 eqcomi ⊢ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) = 𝑋
42 41 fveq2i ⊢ ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) = ( .r ‘ 𝑋 )
43 6 5 40 42 isring ⊢ ( 𝑋 ∈ Ring ↔ ( 𝑋 ∈ Grp ∧ ( mulGrp ‘ 𝑋 ) ∈ Mnd ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ∀ 𝑐 ∈ 𝐵 ( ( 𝑎 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) ( 𝑏 ( +g ‘ 𝑋 ) 𝑐 ) ) = ( ( 𝑎 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑏 ) ( +g ‘ 𝑋 ) ( 𝑎 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑐 ) ) ∧ ( ( 𝑎 ( +g ‘ 𝑋 ) 𝑏 ) ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑐 ) = ( ( 𝑎 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑐 ) ( +g ‘ 𝑋 ) ( 𝑏 ( .r ‘ ( 𝑌 sSet ⟨ ( .r ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ 𝐶 ) ⟩ ) ) 𝑐 ) ) ) ) )
44 39 43 sylnibr ⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐶 ∈ 𝐵 ) → ¬ 𝑋 ∈ Ring )
45 df-nel ⊢ ( 𝑋 ∉ Ring ↔ ¬ 𝑋 ∈ Ring )
46 44 45 sylibr ⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐶 ∈ 𝐵 ) → 𝑋 ∉ Ring )