Theorem precofvallem

Description: Lemma for precofval to enable catlid or catrid . (Contributed by Zhi Wang, 11-Oct-2025)

	Ref	Expression
Hypotheses	precofvallem.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
	precofvallem.b	⊢ 𝐵 = ( Base ‘ 𝐸 )
	precofvallem.1	⊢ 1 = ( Id ‘ 𝐷 )
	precofvallem.i	⊢ 𝐼 = ( Id ‘ 𝐸 )
	precofvallem.f	⊢ ( 𝜑 → 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 )
	precofvallem.k	⊢ ( 𝜑 → 𝐾 ( 𝐷 Func 𝐸 ) 𝐿 )
	precofvallem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
Assertion	precofvallem	⊢ ( 𝜑 → ( ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑋 ) ) ‘ ( ( 1 ∘ 𝐹 ) ‘ 𝑋 ) ) = ( 𝐼 ‘ ( 𝐾 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ∧ ( 𝐾 ‘ ( 𝐹 ‘ 𝑋 ) ) ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		precofvallem.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
2		precofvallem.b	⊢ 𝐵 = ( Base ‘ 𝐸 )
3		precofvallem.1	⊢ 1 = ( Id ‘ 𝐷 )
4		precofvallem.i	⊢ 𝐼 = ( Id ‘ 𝐸 )
5		precofvallem.f	⊢ ( 𝜑 → 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 )
6		precofvallem.k	⊢ ( 𝜑 → 𝐾 ( 𝐷 Func 𝐸 ) 𝐿 )
7		precofvallem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
8		eqid	⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
9	1 8 5	funcf1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐷 ) )
10	9 7	fvco3d	⊢ ( 𝜑 → ( ( 1 ∘ 𝐹 ) ‘ 𝑋 ) = ( 1 ‘ ( 𝐹 ‘ 𝑋 ) ) )
11	10	fveq2d	⊢ ( 𝜑 → ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑋 ) ) ‘ ( ( 1 ∘ 𝐹 ) ‘ 𝑋 ) ) = ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑋 ) ) ‘ ( 1 ‘ ( 𝐹 ‘ 𝑋 ) ) ) )
12	9 7	ffvelcdmd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) )
13	8 3 4 6 12	funcid	⊢ ( 𝜑 → ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑋 ) ) ‘ ( 1 ‘ ( 𝐹 ‘ 𝑋 ) ) ) = ( 𝐼 ‘ ( 𝐾 ‘ ( 𝐹 ‘ 𝑋 ) ) ) )
14	11 13	eqtrd	⊢ ( 𝜑 → ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑋 ) ) ‘ ( ( 1 ∘ 𝐹 ) ‘ 𝑋 ) ) = ( 𝐼 ‘ ( 𝐾 ‘ ( 𝐹 ‘ 𝑋 ) ) ) )
15	8 2 6	funcf1	⊢ ( 𝜑 → 𝐾 : ( Base ‘ 𝐷 ) ⟶ 𝐵 )
16	15 12	ffvelcdmd	⊢ ( 𝜑 → ( 𝐾 ‘ ( 𝐹 ‘ 𝑋 ) ) ∈ 𝐵 )
17	14 16	jca	⊢ ( 𝜑 → ( ( ( ( 𝐹 ‘ 𝑋 ) 𝐿 ( 𝐹 ‘ 𝑋 ) ) ‘ ( ( 1 ∘ 𝐹 ) ‘ 𝑋 ) ) = ( 𝐼 ‘ ( 𝐾 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ∧ ( 𝐾 ‘ ( 𝐹 ‘ 𝑋 ) ) ∈ 𝐵 ) )