Metamath Proof Explorer


Theorem ricsym

Description: Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025)

Ref Expression
Assertion ricsym ( 𝑅𝑟 𝑆𝑆𝑟 𝑅 )

Proof

Step Hyp Ref Expression
1 brric ( 𝑅𝑟 𝑆 ↔ ( 𝑅 RingIso 𝑆 ) ≠ ∅ )
2 n0 ( ( 𝑅 RingIso 𝑆 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 RingIso 𝑆 ) )
3 rimcnv ( 𝑓 ∈ ( 𝑅 RingIso 𝑆 ) → 𝑓 ∈ ( 𝑆 RingIso 𝑅 ) )
4 brrici ( 𝑓 ∈ ( 𝑆 RingIso 𝑅 ) → 𝑆𝑟 𝑅 )
5 3 4 syl ( 𝑓 ∈ ( 𝑅 RingIso 𝑆 ) → 𝑆𝑟 𝑅 )
6 5 exlimiv ( ∃ 𝑓 𝑓 ∈ ( 𝑅 RingIso 𝑆 ) → 𝑆𝑟 𝑅 )
7 2 6 sylbi ( ( 𝑅 RingIso 𝑆 ) ≠ ∅ → 𝑆𝑟 𝑅 )
8 1 7 sylbi ( 𝑅𝑟 𝑆𝑆𝑟 𝑅 )