This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lmicdim

Description: Module isomorphisms preserve vector space dimensions. (Contributed by Thierry Arnoux, 25-Mar-2025)

Ref Expression
Hypotheses lmicdim.1 ⊢ ( 𝜑 → 𝑆 ≃𝑚 𝑇 )
lmicdim.2 ⊢ ( 𝜑 → 𝑆 ∈ LVec )
Assertion lmicdim ( 𝜑 → ( dim ‘ 𝑆 ) = ( dim ‘ 𝑇 ) )

Proof

Step Hyp Ref Expression
1 lmicdim.1 ⊢ ( 𝜑 → 𝑆 ≃𝑚 𝑇 )
2 lmicdim.2 ⊢ ( 𝜑 → 𝑆 ∈ LVec )
3 brlmic ⊢ ( 𝑆 ≃𝑚 𝑇 ↔ ( 𝑆 LMIso 𝑇 ) ≠ ∅ )
4 1 3 sylib ⊢ ( 𝜑 → ( 𝑆 LMIso 𝑇 ) ≠ ∅ )
5 n0 ⊢ ( ( 𝑆 LMIso 𝑇 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝑆 LMIso 𝑇 ) )
6 4 5 sylib ⊢ ( 𝜑 → ∃ 𝑓 𝑓 ∈ ( 𝑆 LMIso 𝑇 ) )
7 simpr ⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑆 LMIso 𝑇 ) ) → 𝑓 ∈ ( 𝑆 LMIso 𝑇 ) )
8 2 adantr ⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑆 LMIso 𝑇 ) ) → 𝑆 ∈ LVec )
9 7 8 lmimdim ⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑆 LMIso 𝑇 ) ) → ( dim ‘ 𝑆 ) = ( dim ‘ 𝑇 ) )
10 6 9 exlimddv ⊢ ( 𝜑 → ( dim ‘ 𝑆 ) = ( dim ‘ 𝑇 ) )