Theorem frmdval

Description: Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015)

	Ref	Expression
Hypotheses	frmdval.m	⊢ 𝑀 = ( freeMnd ‘ 𝐼 )
	frmdval.b	⊢ ( 𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼 )
	frmdval.p	⊢ + = ( ++ ↾ ( 𝐵 × 𝐵 ) )
Assertion	frmdval	⊢ ( 𝐼 ∈ 𝑉 → 𝑀 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ } )

Proof

Step	Hyp	Ref	Expression
1		frmdval.m	⊢ 𝑀 = ( freeMnd ‘ 𝐼 )
2		frmdval.b	⊢ ( 𝐼 ∈ 𝑉 → 𝐵 = Word 𝐼 )
3		frmdval.p	⊢ + = ( ++ ↾ ( 𝐵 × 𝐵 ) )
4		df-frmd	⊢ freeMnd = ( 𝑖 ∈ V ↦ { ⟨ ( Base ‘ ndx ) , Word 𝑖 ⟩ , ⟨ ( +g ‘ ndx ) , ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) ⟩ } )
5		wrdeq	⊢ ( 𝑖 = 𝐼 → Word 𝑖 = Word 𝐼 )
6	2	eqcomd	⊢ ( 𝐼 ∈ 𝑉 → Word 𝐼 = 𝐵 )
7	5 6	sylan9eqr	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → Word 𝑖 = 𝐵 )
8	7	opeq2d	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → ⟨ ( Base ‘ ndx ) , Word 𝑖 ⟩ = ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ )
9	7	sqxpeqd	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → ( Word 𝑖 × Word 𝑖 ) = ( 𝐵 × 𝐵 ) )
10	9	reseq2d	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) = ( ++ ↾ ( 𝐵 × 𝐵 ) ) )
11	10 3	eqtr4di	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) = + )
12	11	opeq2d	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → ⟨ ( +g ‘ ndx ) , ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) ⟩ = ⟨ ( +g ‘ ndx ) , + ⟩ )
13	8 12	preq12d	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑖 = 𝐼 ) → { ⟨ ( Base ‘ ndx ) , Word 𝑖 ⟩ , ⟨ ( +g ‘ ndx ) , ( ++ ↾ ( Word 𝑖 × Word 𝑖 ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ } )
14		elex	⊢ ( 𝐼 ∈ 𝑉 → 𝐼 ∈ V )
15		prex	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ } ∈ V
16	15	a1i	⊢ ( 𝐼 ∈ 𝑉 → { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ } ∈ V )
17	4 13 14 16	fvmptd2	⊢ ( 𝐼 ∈ 𝑉 → ( freeMnd ‘ 𝐼 ) = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ } )
18	1 17	syl5eq	⊢ ( 𝐼 ∈ 𝑉 → 𝑀 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ } )