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Metamath Proof Explorer

Theorem fldgenss

Description: Generated subfields preserve subset ordering. ( see lspss and spanss ) (Contributed by Thierry Arnoux, 15-Jan-2025)

Ref Expression
Hypotheses fldgenval.1 ⊢ 𝐵 = ( Base ‘ 𝐹 )
fldgenval.2 ⊢ ( 𝜑 → 𝐹 ∈ DivRing )
fldgenval.3 ⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
fldgenss.t ⊢ ( 𝜑 → 𝑇 ⊆ 𝑆 )
Assertion fldgenss ( 𝜑 → ( 𝐹 fldGen 𝑇 ) ⊆ ( 𝐹 fldGen 𝑆 ) )

Proof

Step Hyp Ref Expression
1 fldgenval.1 ⊢ 𝐵 = ( Base ‘ 𝐹 )
2 fldgenval.2 ⊢ ( 𝜑 → 𝐹 ∈ DivRing )
3 fldgenval.3 ⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
4 fldgenss.t ⊢ ( 𝜑 → 𝑇 ⊆ 𝑆 )
5 4 adantr ⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑎 ) → 𝑇 ⊆ 𝑆 )
6 simpr ⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑎 ) → 𝑆 ⊆ 𝑎 )
7 5 6 sstrd ⊢ ( ( 𝜑 ∧ 𝑆 ⊆ 𝑎 ) → 𝑇 ⊆ 𝑎 )
8 7 ex ⊢ ( 𝜑 → ( 𝑆 ⊆ 𝑎 → 𝑇 ⊆ 𝑎 ) )
9 8 adantr ⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ) → ( 𝑆 ⊆ 𝑎 → 𝑇 ⊆ 𝑎 ) )
10 9 ss2rabdv ⊢ ( 𝜑 → { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑆 ⊆ 𝑎 } ⊆ { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑇 ⊆ 𝑎 } )
11 intss ⊢ ( { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑆 ⊆ 𝑎 } ⊆ { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑇 ⊆ 𝑎 } → ∩ { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑇 ⊆ 𝑎 } ⊆ ∩ { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑆 ⊆ 𝑎 } )
12 10 11 syl ⊢ ( 𝜑 → ∩ { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑇 ⊆ 𝑎 } ⊆ ∩ { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑆 ⊆ 𝑎 } )
13 4 3 sstrd ⊢ ( 𝜑 → 𝑇 ⊆ 𝐵 )
14 1 2 13 fldgenval ⊢ ( 𝜑 → ( 𝐹 fldGen 𝑇 ) = ∩ { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑇 ⊆ 𝑎 } )
15 1 2 3 fldgenval ⊢ ( 𝜑 → ( 𝐹 fldGen 𝑆 ) = ∩ { 𝑎 ∈ ( SubDRing ‘ 𝐹 ) ∣ 𝑆 ⊆ 𝑎 } )
16 12 14 15 3sstr4d ⊢ ( 𝜑 → ( 𝐹 fldGen 𝑇 ) ⊆ ( 𝐹 fldGen 𝑆 ) )