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Metamath Proof Explorer

Theorem mpofrlmd

Description: Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019) (Proof shortened by AV, 3-Jul-2022)

Ref Expression
Hypotheses mpofrlmd.f ⊢ 𝐹 = ( 𝑅 freeLMod ( 𝑁 × 𝑀 ) )
mpofrlmd.v ⊢ 𝑉 = ( Base ‘ 𝐹 )
mpofrlmd.s ⊢ ( ( 𝑖 = 𝑎 ∧ 𝑗 = 𝑏 ) → 𝐴 = 𝐵 )
mpofrlmd.a ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀 ) → 𝐴 ∈ 𝑋 )
mpofrlmd.b ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀 ) → 𝐵 ∈ 𝑌 )
mpofrlmd.e ⊢ ( 𝜑 → ( 𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) )
Assertion mpofrlmd ( 𝜑 → ( 𝑍 = ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑀 ↦ 𝐵 ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑀 ( 𝑖 𝑍 𝑗 ) = 𝐴 ) )

Proof

Step Hyp Ref Expression
1 mpofrlmd.f ⊢ 𝐹 = ( 𝑅 freeLMod ( 𝑁 × 𝑀 ) )
2 mpofrlmd.v ⊢ 𝑉 = ( Base ‘ 𝐹 )
3 mpofrlmd.s ⊢ ( ( 𝑖 = 𝑎 ∧ 𝑗 = 𝑏 ) → 𝐴 = 𝐵 )
4 mpofrlmd.a ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀 ) → 𝐴 ∈ 𝑋 )
5 mpofrlmd.b ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀 ) → 𝐵 ∈ 𝑌 )
6 mpofrlmd.e ⊢ ( 𝜑 → ( 𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) )
7 xpexg ⊢ ( ( 𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ) → ( 𝑁 × 𝑀 ) ∈ V )
8 7 anim1i ⊢ ( ( ( 𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ) ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝑁 × 𝑀 ) ∈ V ∧ 𝑍 ∈ 𝑉 ) )
9 8 3impa ⊢ ( ( 𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝑁 × 𝑀 ) ∈ V ∧ 𝑍 ∈ 𝑉 ) )
10 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
11 1 10 2 frlmbasf ⊢ ( ( ( 𝑁 × 𝑀 ) ∈ V ∧ 𝑍 ∈ 𝑉 ) → 𝑍 : ( 𝑁 × 𝑀 ) ⟶ ( Base ‘ 𝑅 ) )
12 ffn ⊢ ( 𝑍 : ( 𝑁 × 𝑀 ) ⟶ ( Base ‘ 𝑅 ) → 𝑍 Fn ( 𝑁 × 𝑀 ) )
13 6 9 11 12 4syl ⊢ ( 𝜑 → 𝑍 Fn ( 𝑁 × 𝑀 ) )
14 13 3 4 5 fnmpoovd ⊢ ( 𝜑 → ( 𝑍 = ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑀 ↦ 𝐵 ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑀 ( 𝑖 𝑍 𝑗 ) = 𝐴 ) )