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Metamath Proof Explorer

Theorem gsummulglem

Description: Lemma for gsummulg and gsummulgz . (Contributed by Mario Carneiro, 7-Jan-2015) (Revised by AV, 6-Jun-2019)

Ref Expression
Hypotheses gsummulg.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
gsummulg.z ⊢ 0 = ( 0g ‘ 𝐺 )
gsummulg.t ⊢ · = ( .g ‘ 𝐺 )
gsummulg.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
gsummulg.f ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
gsummulg.w ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
gsummulglem.g ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
gsummulglem.n ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
gsummulglem.o ⊢ ( 𝜑 → ( 𝐺 ∈ Abel ∨ 𝑁 ∈ ℕ0 ) )
Assertion gsummulglem ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑁 · 𝑋 ) ) ) = ( 𝑁 · ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )

Proof

Step Hyp Ref Expression
1 gsummulg.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 gsummulg.z ⊢ 0 = ( 0g ‘ 𝐺 )
3 gsummulg.t ⊢ · = ( .g ‘ 𝐺 )
4 gsummulg.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5 gsummulg.f ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
6 gsummulg.w ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
7 gsummulglem.g ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
8 gsummulglem.n ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
9 gsummulglem.o ⊢ ( 𝜑 → ( 𝐺 ∈ Abel ∨ 𝑁 ∈ ℕ0 ) )
10 cmnmnd ⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd )
11 7 10 syl ⊢ ( 𝜑 → 𝐺 ∈ Mnd )
12 1 3 mulgghm ⊢ ( ( 𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 GrpHom 𝐺 ) )
13 ghmmhm ⊢ ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 GrpHom 𝐺 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) )
14 12 13 syl ⊢ ( ( 𝐺 ∈ Abel ∧ 𝑁 ∈ ℤ ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) )
15 14 expcom ⊢ ( 𝑁 ∈ ℤ → ( 𝐺 ∈ Abel → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) ) )
16 8 15 syl ⊢ ( 𝜑 → ( 𝐺 ∈ Abel → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) ) )
17 1 3 mulgmhm ⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) )
18 17 ex ⊢ ( 𝐺 ∈ CMnd → ( 𝑁 ∈ ℕ0 → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) ) )
19 7 18 syl ⊢ ( 𝜑 → ( 𝑁 ∈ ℕ0 → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) ) )
20 16 19 9 mpjaod ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 · 𝑥 ) ) ∈ ( 𝐺 MndHom 𝐺 ) )
21 oveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝑁 · 𝑥 ) = ( 𝑁 · 𝑋 ) )
22 oveq2 ⊢ ( 𝑥 = ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) → ( 𝑁 · 𝑥 ) = ( 𝑁 · ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )
23 1 2 7 11 4 20 5 6 21 22 gsummhm2 ⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ ( 𝑁 · 𝑋 ) ) ) = ( 𝑁 · ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) ) )