Metamath Proof Explorer


Theorem ricsym

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

Ref Expression
Assertion ricsym R 𝑟 S S 𝑟 R

Proof

Step Hyp Ref Expression
1 brric R 𝑟 S R RingIso S
2 n0 R RingIso S f f R RingIso S
3 rimcnv f R RingIso S f -1 S RingIso R
4 brrici f -1 S RingIso R S 𝑟 R
5 3 4 syl f R RingIso S S 𝑟 R
6 5 exlimiv f f R RingIso S S 𝑟 R
7 2 6 sylbi R RingIso S S 𝑟 R
8 1 7 sylbi R 𝑟 S S 𝑟 R