Description: Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | lmiclcl | ⊢ ( 𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brlmic | ⊢ ( 𝑅 ≃𝑚 𝑆 ↔ ( 𝑅 LMIso 𝑆 ) ≠ ∅ ) | |
2 | n0 | ⊢ ( ( 𝑅 LMIso 𝑆 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) ) | |
3 | 1 2 | bitri | ⊢ ( 𝑅 ≃𝑚 𝑆 ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) ) |
4 | lmimlmhm | ⊢ ( 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) → 𝑓 ∈ ( 𝑅 LMHom 𝑆 ) ) | |
5 | lmhmlmod1 | ⊢ ( 𝑓 ∈ ( 𝑅 LMHom 𝑆 ) → 𝑅 ∈ LMod ) | |
6 | 4 5 | syl | ⊢ ( 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) → 𝑅 ∈ LMod ) |
7 | 6 | exlimiv | ⊢ ( ∃ 𝑓 𝑓 ∈ ( 𝑅 LMIso 𝑆 ) → 𝑅 ∈ LMod ) |
8 | 3 7 | sylbi | ⊢ ( 𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod ) |