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

Metamath Proof Explorer

Theorem gictr

Description: Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Assertion gictr ( ( 𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇 ) → 𝑅 ≃𝑔 𝑇 )

Proof

Step Hyp Ref Expression
1 brgic ⊢ ( 𝑅 ≃𝑔 𝑆 ↔ ( 𝑅 GrpIso 𝑆 ) ≠ ∅ )
2 brgic ⊢ ( 𝑆 ≃𝑔 𝑇 ↔ ( 𝑆 GrpIso 𝑇 ) ≠ ∅ )
3 n0 ⊢ ( ( 𝑅 GrpIso 𝑆 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) )
4 n0 ⊢ ( ( 𝑆 GrpIso 𝑇 ) ≠ ∅ ↔ ∃ 𝑔 𝑔 ∈ ( 𝑆 GrpIso 𝑇 ) )
5 exdistrv ⊢ ( ∃ 𝑓 ∃ 𝑔 ( 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) ∧ 𝑔 ∈ ( 𝑆 GrpIso 𝑇 ) ) ↔ ( ∃ 𝑓 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) ∧ ∃ 𝑔 𝑔 ∈ ( 𝑆 GrpIso 𝑇 ) ) )
6 gimco ⊢ ( ( 𝑔 ∈ ( 𝑆 GrpIso 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) ) → ( 𝑔 ∘ 𝑓 ) ∈ ( 𝑅 GrpIso 𝑇 ) )
7 brgici ⊢ ( ( 𝑔 ∘ 𝑓 ) ∈ ( 𝑅 GrpIso 𝑇 ) → 𝑅 ≃𝑔 𝑇 )
8 6 7 syl ⊢ ( ( 𝑔 ∈ ( 𝑆 GrpIso 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) ) → 𝑅 ≃𝑔 𝑇 )
9 8 ancoms ⊢ ( ( 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) ∧ 𝑔 ∈ ( 𝑆 GrpIso 𝑇 ) ) → 𝑅 ≃𝑔 𝑇 )
10 9 exlimivv ⊢ ( ∃ 𝑓 ∃ 𝑔 ( 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) ∧ 𝑔 ∈ ( 𝑆 GrpIso 𝑇 ) ) → 𝑅 ≃𝑔 𝑇 )
11 5 10 sylbir ⊢ ( ( ∃ 𝑓 𝑓 ∈ ( 𝑅 GrpIso 𝑆 ) ∧ ∃ 𝑔 𝑔 ∈ ( 𝑆 GrpIso 𝑇 ) ) → 𝑅 ≃𝑔 𝑇 )
12 3 4 11 syl2anb ⊢ ( ( ( 𝑅 GrpIso 𝑆 ) ≠ ∅ ∧ ( 𝑆 GrpIso 𝑇 ) ≠ ∅ ) → 𝑅 ≃𝑔 𝑇 )
13 1 2 12 syl2anb ⊢ ( ( 𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇 ) → 𝑅 ≃𝑔 𝑇 )