Metamath Proof Explorer


Theorem scandxnbasendx

Description: The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024)

Ref Expression
Assertion scandxnbasendx
|- ( Scalar ` ndx ) =/= ( Base ` ndx )

Proof

Step Hyp Ref Expression
1 1re
 |-  1 e. RR
2 1lt5
 |-  1 < 5
3 1 2 gtneii
 |-  5 =/= 1
4 scandx
 |-  ( Scalar ` ndx ) = 5
5 basendx
 |-  ( Base ` ndx ) = 1
6 4 5 neeq12i
 |-  ( ( Scalar ` ndx ) =/= ( Base ` ndx ) <-> 5 =/= 1 )
7 3 6 mpbir
 |-  ( Scalar ` ndx ) =/= ( Base ` ndx )