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

Metamath Proof Explorer

Theorem fldgenid

Description: The subfield of a field F generated by the whole base set of F is F itself. (Contributed by Thierry Arnoux, 11-Jan-2025)

Ref Expression
Hypotheses fldgenval.1 ⊢ 𝐵 = ( Base ‘ 𝐹 )
fldgenval.2 ⊢ ( 𝜑 → 𝐹 ∈ DivRing )
Assertion fldgenid ( 𝜑 → ( 𝐹 fldGen 𝐵 ) = 𝐵 )

Proof

Step Hyp Ref Expression
1 fldgenval.1 ⊢ 𝐵 = ( Base ‘ 𝐹 )
2 fldgenval.2 ⊢ ( 𝜑 → 𝐹 ∈ DivRing )
3 ssidd ⊢ ( 𝜑 → 𝐵 ⊆ 𝐵 )
4 1 2 3 fldgenssv ⊢ ( 𝜑 → ( 𝐹 fldGen 𝐵 ) ⊆ 𝐵 )
5 1 2 3 fldgenssid ⊢ ( 𝜑 → 𝐵 ⊆ ( 𝐹 fldGen 𝐵 ) )
6 4 5 eqssd ⊢ ( 𝜑 → ( 𝐹 fldGen 𝐵 ) = 𝐵 )