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Metamath Proof Explorer

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 3 grpmndd ⊢ ( 𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd )
5 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
6 2 5 gsumccatsn ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑊 ++ ⟨“ 𝑍 ”⟩ ) ) = ( ( 𝐺 Σg 𝑊 ) ( +g ‘ 𝐺 ) 𝑍 ) )
7 4 6 syl3an1 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑊 ++ ⟨“ 𝑍 ”⟩ ) ) = ( ( 𝐺 Σg 𝑊 ) ( +g ‘ 𝐺 ) 𝑍 ) )
8 4 3ad2ant1 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝐺 ∈ Mnd )
9 simp2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝑊 ∈ Word 𝐵 )
10 2 gsumwcl ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ) → ( 𝐺 Σg 𝑊 ) ∈ 𝐵 )
11 8 9 10 syl2anc ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg 𝑊 ) ∈ 𝐵 )
12 simp3 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝑍 ∈ 𝐵 )
13 1 2 5 symgov ⊢ ( ( ( 𝐺 Σg 𝑊 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝐺 Σg 𝑊 ) ( +g ‘ 𝐺 ) 𝑍 ) = ( ( 𝐺 Σg 𝑊 ) ∘ 𝑍 ) )
14 11 12 13 syl2anc ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝐺 Σg 𝑊 ) ( +g ‘ 𝐺 ) 𝑍 ) = ( ( 𝐺 Σg 𝑊 ) ∘ 𝑍 ) )
15 7 14 eqtrd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑊 ++ ⟨“ 𝑍 ”⟩ ) ) = ( ( 𝐺 Σg 𝑊 ) ∘ 𝑍 ) )