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Metamath Proof Explorer

Theorem ipsstr

Description: Lemma to shorten proofs of ipsbase through ipsvsca . (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

Ref Expression
Hypothesis ipspart.a 
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )
Assertion ipsstr 
|- A Struct <. 1 , 8 >.

Proof

Step Hyp Ref Expression
1 ipspart.a 
 |-  A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )
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 6lt8 
 |-  6 < 8
10 8nn 
 |-  8 e. NN
11 ipndx 
 |-  ( .i ` ndx ) = 8
12 4 5 6 7 8 9 10 11 strle3 
 |-  { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } Struct <. 5 , 8 >.
13 3lt5 
 |-  3 < 5
14 3 12 13 strleun 
 |-  ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } ) Struct <. 1 , 8 >.
15 1 14 eqbrtri 
 |-  A Struct <. 1 , 8 >.