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 ) )