Metamath Proof Explorer

Theorem algstr

Description: Lemma to shorten proofs of algbase through algvsca . (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis algpart.a 
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } )
Assertion algstr 
|- A Struct <. 1 , 6 >.

Proof

Step Hyp Ref Expression
1 algpart.a 
 |-  A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } )
2 eqid 
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. }
3 2 rngstr 
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } Struct <. 1 , 3 >.
4 5nn 
 |-  5 e. NN
5 scandx 
 |-  ( Scalar ` ndx ) = 5
6 5lt6 
 |-  5 < 6
7 6nn 
 |-  6 e. NN
8 vscandx 
 |-  ( .s ` ndx ) = 6
9 4 5 6 7 8 strle2 
 |-  { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } Struct <. 5 , 6 >.
10 3lt5 
 |-  3 < 5
11 3 9 10 strleun 
 |-  ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) Struct <. 1 , 6 >.
12 1 11 eqbrtri 
 |-  A Struct <. 1 , 6 >.