Metamath Proof Explorer


Theorem lindsmm2

Description: The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015)

Ref Expression
Hypotheses lindfmm.b 𝐵 = ( Base ‘ 𝑆 )
lindfmm.c 𝐶 = ( Base ‘ 𝑇 )
Assertion lindsmm2 ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵1-1𝐶𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → ( 𝐺𝐹 ) ∈ ( LIndS ‘ 𝑇 ) )

Proof

Step Hyp Ref Expression
1 lindfmm.b 𝐵 = ( Base ‘ 𝑆 )
2 lindfmm.c 𝐶 = ( Base ‘ 𝑇 )
3 simp3 ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵1-1𝐶𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → 𝐹 ∈ ( LIndS ‘ 𝑆 ) )
4 1 linds1 ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) → 𝐹𝐵 )
5 1 2 lindsmm ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵1-1𝐶𝐹𝐵 ) → ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) ↔ ( 𝐺𝐹 ) ∈ ( LIndS ‘ 𝑇 ) ) )
6 4 5 syl3an3 ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵1-1𝐶𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) ↔ ( 𝐺𝐹 ) ∈ ( LIndS ‘ 𝑇 ) ) )
7 3 6 mpbid ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵1-1𝐶𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → ( 𝐺𝐹 ) ∈ ( LIndS ‘ 𝑇 ) )