Step |
Hyp |
Ref |
Expression |
1 |
|
islmhmd.x |
⊢ 𝑋 = ( Base ‘ 𝑆 ) |
2 |
|
islmhmd.a |
⊢ · = ( ·𝑠 ‘ 𝑆 ) |
3 |
|
islmhmd.b |
⊢ × = ( ·𝑠 ‘ 𝑇 ) |
4 |
|
islmhmd.k |
⊢ 𝐾 = ( Scalar ‘ 𝑆 ) |
5 |
|
islmhmd.j |
⊢ 𝐽 = ( Scalar ‘ 𝑇 ) |
6 |
|
islmhmd.n |
⊢ 𝑁 = ( Base ‘ 𝐾 ) |
7 |
|
islmhmd.s |
⊢ ( 𝜑 → 𝑆 ∈ LMod ) |
8 |
|
islmhmd.t |
⊢ ( 𝜑 → 𝑇 ∈ LMod ) |
9 |
|
islmhmd.c |
⊢ ( 𝜑 → 𝐽 = 𝐾 ) |
10 |
|
islmhmd.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
11 |
|
islmhmd.l |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝑥 × ( 𝐹 ‘ 𝑦 ) ) ) |
12 |
11
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑁 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝑥 × ( 𝐹 ‘ 𝑦 ) ) ) |
13 |
10 9 12
|
3jca |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐽 = 𝐾 ∧ ∀ 𝑥 ∈ 𝑁 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝑥 × ( 𝐹 ‘ 𝑦 ) ) ) ) |
14 |
4 5 6 1 2 3
|
islmhm |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ↔ ( ( 𝑆 ∈ LMod ∧ 𝑇 ∈ LMod ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝐽 = 𝐾 ∧ ∀ 𝑥 ∈ 𝑁 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝑥 × ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
15 |
7 8 13 14
|
syl21anbrc |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |