Step |
Hyp |
Ref |
Expression |
1 |
|
lindfmm.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
2 |
|
lindfmm.c |
⊢ 𝐶 = ( Base ‘ 𝑇 ) |
3 |
|
simp3 |
⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → 𝐹 ∈ ( LIndS ‘ 𝑆 ) ) |
4 |
1
|
linds1 |
⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) → 𝐹 ⊆ 𝐵 ) |
5 |
1 2
|
lindsmm |
⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ⊆ 𝐵 ) → ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) ↔ ( 𝐺 “ 𝐹 ) ∈ ( LIndS ‘ 𝑇 ) ) ) |
6 |
4 5
|
syl3an3 |
⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) ↔ ( 𝐺 “ 𝐹 ) ∈ ( LIndS ‘ 𝑇 ) ) ) |
7 |
3 6
|
mpbid |
⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → ( 𝐺 “ 𝐹 ) ∈ ( LIndS ‘ 𝑇 ) ) |