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Metamath Proof Explorer

Theorem mvrval2

Description: Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015)

Ref Expression
Hypotheses mvrfval.v ⊢ 𝑉 = ( 𝐼 mVar 𝑅 )
mvrfval.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
mvrfval.z ⊢ 0 = ( 0g ‘ 𝑅 )
mvrfval.o ⊢ 1 = ( 1r ‘ 𝑅 )
mvrfval.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
mvrfval.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑌 )
mvrval.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
mvrval2.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
Assertion mvrval2 ( 𝜑 → ( ( 𝑉 ‘ 𝑋 ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )

Proof

Step Hyp Ref Expression
1 mvrfval.v ⊢ 𝑉 = ( 𝐼 mVar 𝑅 )
2 mvrfval.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
3 mvrfval.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 mvrfval.o ⊢ 1 = ( 1r ‘ 𝑅 )
5 mvrfval.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
6 mvrfval.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑌 )
7 mvrval.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
8 mvrval2.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
9 1 2 3 4 5 6 7 mvrval ⊢ ( 𝜑 → ( 𝑉 ‘ 𝑋 ) = ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) )
10 9 fveq1d ⊢ ( 𝜑 → ( ( 𝑉 ‘ 𝑋 ) ‘ 𝐹 ) = ( ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) )
11 eqeq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ↔ 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
12 11 ifbid ⊢ ( 𝑓 = 𝐹 → if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )
13 eqid ⊢ ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) = ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )
14 4 fvexi ⊢ 1 ∈ V
15 3 fvexi ⊢ 0 ∈ V
16 14 15 ifex ⊢ if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ∈ V
17 12 13 16 fvmpt ⊢ ( 𝐹 ∈ 𝐷 → ( ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )
18 8 17 syl ⊢ ( 𝜑 → ( ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )
19 10 18 eqtrd ⊢ ( 𝜑 → ( ( 𝑉 ‘ 𝑋 ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )