Metamath Proof Explorer


Theorem rspsnid

Description: A principal ideal contains the element that generates it. (Contributed by Thierry Arnoux, 15-Jan-2024)

Ref Expression
Hypotheses rspsnid.b B = Base R
rspsnid.k K = RSpan R
Assertion rspsnid R Ring G B G K G

Proof

Step Hyp Ref Expression
1 rspsnid.b B = Base R
2 rspsnid.k K = RSpan R
3 snssi G B G B
4 2 1 rspssid R Ring G B G K G
5 3 4 sylan2 R Ring G B G K G
6 snssg G B G K G G K G
7 6 adantl R Ring G B G K G G K G
8 5 7 mpbird R Ring G B G K G