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 ) )