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Metamath Proof Explorer

Theorem mulgghm

Description: The map from x to n x for a fixed integer n is a group homomorphism if the group is commutative. (Contributed by Mario Carneiro, 4-May-2015)

Ref Expression
Hypotheses mulgmhm.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
mulgmhm.m ⊢ · = ( .g ‘ 𝐺 )
Assertion mulgghm ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ∈ ( 𝐺 GrpHom 𝐺 ) )

Proof

Step Hyp Ref Expression
1 mulgmhm.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 mulgmhm.m ⊢ · = ( .g ‘ 𝐺 )
3 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
4 ablgrp ⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
5 4 adantr ⊢ ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) → 𝐺 ∈ Grp )
6 1 2 mulgcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝐵 ) → ( 𝑀 · 𝑥 ) ∈ 𝐵 )
7 4 6 syl3an1 ⊢ ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝐵 ) → ( 𝑀 · 𝑥 ) ∈ 𝐵 )
8 7 3expa ⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑀 · 𝑥 ) ∈ 𝐵 )
9 8 fmpttd ⊢ ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) : 𝐵 ⟶ 𝐵 )
10 3anass ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ↔ ( 𝑀 ∈ ℤ ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) )
11 1 2 3 mulgdi ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) )
12 10 11 sylan2br ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑀 ∈ ℤ ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ) → ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) )
13 12 anassrs ⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) )
14 1 3 grpcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝐵 )
15 14 3expb ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝐵 )
16 5 15 sylan ⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝐵 )
17 oveq2 ⊢ ( 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ( 𝑀 · 𝑥 ) = ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) )
18 eqid ⊢ ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) = ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) )
19 ovex ⊢ ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ V
20 17 18 19 fvmpt ⊢ ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) )
21 16 20 syl ⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( 𝑀 · ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) )
22 oveq2 ⊢ ( 𝑥 = 𝑦 → ( 𝑀 · 𝑥 ) = ( 𝑀 · 𝑦 ) )
23 ovex ⊢ ( 𝑀 · 𝑦 ) ∈ V
24 22 18 23 fvmpt ⊢ ( 𝑦 ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) = ( 𝑀 · 𝑦 ) )
25 oveq2 ⊢ ( 𝑥 = 𝑧 → ( 𝑀 · 𝑥 ) = ( 𝑀 · 𝑧 ) )
26 ovex ⊢ ( 𝑀 · 𝑧 ) ∈ V
27 25 18 26 fvmpt ⊢ ( 𝑧 ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) = ( 𝑀 · 𝑧 ) )
28 24 27 oveqan12d ⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) )
29 28 adantl ⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) = ( ( 𝑀 · 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑀 · 𝑧 ) ) )
30 13 21 29 3eqtr4d ⊢ ( ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = ( ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ‘ 𝑧 ) ) )
31 1 1 3 3 5 5 9 30 isghmd ⊢ ( ( 𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑀 · 𝑥 ) ) ∈ ( 𝐺 GrpHom 𝐺 ) )