Theorem gsumccatsymgsn

Description: Homomorphic property of composites of permutations with a singleton. (Contributed by AV, 20-Jan-2019)

	Ref	Expression
Hypotheses	gsumccatsymgsn.g	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
	gsumccatsymgsn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	gsumccatsymgsn	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑊 ++ ⟨“ 𝑍 ”⟩ ) ) = ( ( 𝐺 Σg 𝑊 ) ∘ 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		gsumccatsymgsn.g	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2		gsumccatsymgsn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
3	1	symggrp	⊢ ( 𝐴 ∈ 𝑉 → 𝐺 ∈ Grp )
4		grpmnd	⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
5	3 4	syl	⊢ ( 𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd )
6		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7	2 6	gsumccatsn	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑊 ++ ⟨“ 𝑍 ”⟩ ) ) = ( ( 𝐺 Σg 𝑊 ) ( +g ‘ 𝐺 ) 𝑍 ) )
8	5 7	syl3an1	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑊 ++ ⟨“ 𝑍 ”⟩ ) ) = ( ( 𝐺 Σg 𝑊 ) ( +g ‘ 𝐺 ) 𝑍 ) )
9	5	3ad2ant1	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝐺 ∈ Mnd )
10		simp2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝑊 ∈ Word 𝐵 )
11	2	gsumwcl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ) → ( 𝐺 Σg 𝑊 ) ∈ 𝐵 )
12	9 10 11	syl2anc	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg 𝑊 ) ∈ 𝐵 )
13		simp3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝑍 ∈ 𝐵 )
14	1 2 6	symgov	⊢ ( ( ( 𝐺 Σg 𝑊 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝐺 Σg 𝑊 ) ( +g ‘ 𝐺 ) 𝑍 ) = ( ( 𝐺 Σg 𝑊 ) ∘ 𝑍 ) )
15	12 13 14	syl2anc	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝐺 Σg 𝑊 ) ( +g ‘ 𝐺 ) 𝑍 ) = ( ( 𝐺 Σg 𝑊 ) ∘ 𝑍 ) )
16	8 15	eqtrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑊 ++ ⟨“ 𝑍 ”⟩ ) ) = ( ( 𝐺 Σg 𝑊 ) ∘ 𝑍 ) )