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Metamath Proof Explorer

Theorem dvdsrpropd

Description: The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014)

Ref Expression
Hypotheses rngidpropd.1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
rngidpropd.2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
rngidpropd.3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) )
Assertion dvdsrpropd ( 𝜑 → ( ∥r ‘ 𝐾 ) = ( ∥r ‘ 𝐿 ) )

Proof

Step Hyp Ref Expression
1 rngidpropd.1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
2 rngidpropd.2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
3 rngidpropd.3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) )
4 3 anassrs ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) )
5 4 eqeq1d ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ↔ ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) )
6 5 an32s ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ↔ ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) )
7 6 rexbidva ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ↔ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) )
8 7 pm5.32da ⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ) ↔ ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) ) )
9 1 eleq2d ⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ( Base ‘ 𝐾 ) ) )
10 1 rexeqdv ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ↔ ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ) )
11 9 10 anbi12d ⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ) ↔ ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ) ) )
12 2 eleq2d ⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ( Base ‘ 𝐿 ) ) )
13 2 rexeqdv ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ↔ ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) )
14 12 13 anbi12d ⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) ↔ ( 𝑦 ∈ ( Base ‘ 𝐿 ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) ) )
15 8 11 14 3bitr3d ⊢ ( 𝜑 → ( ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ) ↔ ( 𝑦 ∈ ( Base ‘ 𝐿 ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) ) )
16 15 opabbidv ⊢ ( 𝜑 → { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ) } = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ ( Base ‘ 𝐿 ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) } )
17 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
18 eqid ⊢ ( ∥r ‘ 𝐾 ) = ( ∥r ‘ 𝐾 )
19 eqid ⊢ ( .r ‘ 𝐾 ) = ( .r ‘ 𝐾 )
20 17 18 19 dvdsrval ⊢ ( ∥r ‘ 𝐾 ) = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = 𝑧 ) }
21 eqid ⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
22 eqid ⊢ ( ∥r ‘ 𝐿 ) = ( ∥r ‘ 𝐿 )
23 eqid ⊢ ( .r ‘ 𝐿 ) = ( .r ‘ 𝐿 )
24 21 22 23 dvdsrval ⊢ ( ∥r ‘ 𝐿 ) = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ ( Base ‘ 𝐿 ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = 𝑧 ) }
25 16 20 24 3eqtr4g ⊢ ( 𝜑 → ( ∥r ‘ 𝐾 ) = ( ∥r ‘ 𝐿 ) )