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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 
|- ( ph -> F e. DivRing )
fldgenval.3 
|- ( ph -> S C_ B )
fldgenss.t 
|- ( ph -> T C_ S )
Assertion fldgenss 
|- ( ph -> ( F fldGen T ) C_ ( F fldGen S ) )

Proof

Step Hyp Ref Expression
1 fldgenval.1 
 |-  B = ( Base ` F )
2 fldgenval.2 
 |-  ( ph -> F e. DivRing )
3 fldgenval.3 
 |-  ( ph -> S C_ B )
4 fldgenss.t 
 |-  ( ph -> T C_ S )
5 4 adantr 
 |-  ( ( ph /\ S C_ a ) -> T C_ S )
6 simpr 
 |-  ( ( ph /\ S C_ a ) -> S C_ a )
7 5 6 sstrd 
 |-  ( ( ph /\ S C_ a ) -> T C_ a )
8 7 ex 
 |-  ( ph -> ( S C_ a -> T C_ a ) )
9 8 adantr 
 |-  ( ( ph /\ a e. ( SubDRing ` F ) ) -> ( S C_ a -> T C_ a ) )
10 9 ss2rabdv 
 |-  ( ph -> { a e. ( SubDRing ` F ) | S C_ a } C_ { a e. ( SubDRing ` F ) | T C_ a } )
11 intss 
 |-  ( { a e. ( SubDRing ` F ) | S C_ a } C_ { a e. ( SubDRing ` F ) | T C_ a } -> |^| { a e. ( SubDRing ` F ) | T C_ a } C_ |^| { a e. ( SubDRing ` F ) | S C_ a } )
12 10 11 syl 
 |-  ( ph -> |^| { a e. ( SubDRing ` F ) | T C_ a } C_ |^| { a e. ( SubDRing ` F ) | S C_ a } )
13 4 3 sstrd 
 |-  ( ph -> T C_ B )
14 1 2 13 fldgenval 
 |-  ( ph -> ( F fldGen T ) = |^| { a e. ( SubDRing ` F ) | T C_ a } )
15 1 2 3 fldgenval 
 |-  ( ph -> ( F fldGen S ) = |^| { a e. ( SubDRing ` F ) | S C_ a } )
16 12 14 15 3sstr4d 
 |-  ( ph -> ( F fldGen T ) C_ ( F fldGen S ) )