Theorem mhmvlin

Description: Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015)

	Ref	Expression
Hypotheses	mhmvlin.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
	mhmvlin.p	⊢ + = ( +g ‘ 𝑀 )
	mhmvlin.q	⊢ ⨣ = ( +g ‘ 𝑁 )
Assertion	mhmvlin	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝐹 ∘ ( 𝑋 ∘f + 𝑌 ) ) = ( ( 𝐹 ∘ 𝑋 ) ∘f ⨣ ( 𝐹 ∘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mhmvlin.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
2		mhmvlin.p	⊢ + = ( +g ‘ 𝑀 )
3		mhmvlin.q	⊢ ⨣ = ( +g ‘ 𝑁 )
4		simpl1	⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐼 ) → 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) )
5		elmapi	⊢ ( 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) → 𝑋 : 𝐼 ⟶ 𝐵 )
6	5	3ad2ant2	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑋 : 𝐼 ⟶ 𝐵 )
7	6	ffvelrnda	⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑦 ) ∈ 𝐵 )
8		elmapi	⊢ ( 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) → 𝑌 : 𝐼 ⟶ 𝐵 )
9	8	3ad2ant3	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑌 : 𝐼 ⟶ 𝐵 )
10	9	ffvelrnda	⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑦 ) ∈ 𝐵 )
11	1 2 3	mhmlin	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ ( 𝑋 ‘ 𝑦 ) ∈ 𝐵 ∧ ( 𝑌 ‘ 𝑦 ) ∈ 𝐵 ) → ( 𝐹 ‘ ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) ) = ( ( 𝐹 ‘ ( 𝑋 ‘ 𝑦 ) ) ⨣ ( 𝐹 ‘ ( 𝑌 ‘ 𝑦 ) ) ) )
12	4 7 10 11	syl3anc	⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) ) = ( ( 𝐹 ‘ ( 𝑋 ‘ 𝑦 ) ) ⨣ ( 𝐹 ‘ ( 𝑌 ‘ 𝑦 ) ) ) )
13	12	mpteq2dva	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑦 ∈ 𝐼 ↦ ( 𝐹 ‘ ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) ) ) = ( 𝑦 ∈ 𝐼 ↦ ( ( 𝐹 ‘ ( 𝑋 ‘ 𝑦 ) ) ⨣ ( 𝐹 ‘ ( 𝑌 ‘ 𝑦 ) ) ) ) )
14		mhmrcl1	⊢ ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) → 𝑀 ∈ Mnd )
15	14	adantr	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑦 ∈ 𝐼 ) → 𝑀 ∈ Mnd )
16	15	3ad2antl1	⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐼 ) → 𝑀 ∈ Mnd )
17	1 2	mndcl	⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑋 ‘ 𝑦 ) ∈ 𝐵 ∧ ( 𝑌 ‘ 𝑦 ) ∈ 𝐵 ) → ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) ∈ 𝐵 )
18	16 7 10 17	syl3anc	⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐼 ) → ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) ∈ 𝐵 )
19		elmapex	⊢ ( 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) → ( 𝐵 ∈ V ∧ 𝐼 ∈ V ) )
20	19	simprd	⊢ ( 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) → 𝐼 ∈ V )
21	20	3ad2ant3	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝐼 ∈ V )
22	6	feqmptd	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑋 = ( 𝑦 ∈ 𝐼 ↦ ( 𝑋 ‘ 𝑦 ) ) )
23	9	feqmptd	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝑌 = ( 𝑦 ∈ 𝐼 ↦ ( 𝑌 ‘ 𝑦 ) ) )
24	21 7 10 22 23	offval2	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝑋 ∘f + 𝑌 ) = ( 𝑦 ∈ 𝐼 ↦ ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) ) )
25		eqid	⊢ ( Base ‘ 𝑁 ) = ( Base ‘ 𝑁 )
26	1 25	mhmf	⊢ ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) → 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑁 ) )
27	26	3ad2ant1	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑁 ) )
28	27	feqmptd	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → 𝐹 = ( 𝑧 ∈ 𝐵 ↦ ( 𝐹 ‘ 𝑧 ) ) )
29		fveq2	⊢ ( 𝑧 = ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) ) )
30	18 24 28 29	fmptco	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝐹 ∘ ( 𝑋 ∘f + 𝑌 ) ) = ( 𝑦 ∈ 𝐼 ↦ ( 𝐹 ‘ ( ( 𝑋 ‘ 𝑦 ) + ( 𝑌 ‘ 𝑦 ) ) ) ) )
31		fvexd	⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ ( 𝑋 ‘ 𝑦 ) ) ∈ V )
32		fvexd	⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ ( 𝑌 ‘ 𝑦 ) ) ∈ V )
33		fcompt	⊢ ( ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑁 ) ∧ 𝑋 : 𝐼 ⟶ 𝐵 ) → ( 𝐹 ∘ 𝑋 ) = ( 𝑦 ∈ 𝐼 ↦ ( 𝐹 ‘ ( 𝑋 ‘ 𝑦 ) ) ) )
34	27 6 33	syl2anc	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝐹 ∘ 𝑋 ) = ( 𝑦 ∈ 𝐼 ↦ ( 𝐹 ‘ ( 𝑋 ‘ 𝑦 ) ) ) )
35		fcompt	⊢ ( ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑁 ) ∧ 𝑌 : 𝐼 ⟶ 𝐵 ) → ( 𝐹 ∘ 𝑌 ) = ( 𝑦 ∈ 𝐼 ↦ ( 𝐹 ‘ ( 𝑌 ‘ 𝑦 ) ) ) )
36	27 9 35	syl2anc	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝐹 ∘ 𝑌 ) = ( 𝑦 ∈ 𝐼 ↦ ( 𝐹 ‘ ( 𝑌 ‘ 𝑦 ) ) ) )
37	21 31 32 34 36	offval2	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( ( 𝐹 ∘ 𝑋 ) ∘f ⨣ ( 𝐹 ∘ 𝑌 ) ) = ( 𝑦 ∈ 𝐼 ↦ ( ( 𝐹 ‘ ( 𝑋 ‘ 𝑦 ) ) ⨣ ( 𝐹 ‘ ( 𝑌 ‘ 𝑦 ) ) ) ) )
38	13 30 37	3eqtr4d	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( 𝐵 ↑m 𝐼 ) ∧ 𝑌 ∈ ( 𝐵 ↑m 𝐼 ) ) → ( 𝐹 ∘ ( 𝑋 ∘f + 𝑌 ) ) = ( ( 𝐹 ∘ 𝑋 ) ∘f ⨣ ( 𝐹 ∘ 𝑌 ) ) )