Metamath Proof Explorer

Theorem ipndxnplusgndx

Description: The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024)

Ref Expression
Assertion ipndxnplusgndx 
|- ( .i ` ndx ) =/= ( +g ` ndx )

Proof

Step Hyp Ref Expression
1 2re 
 |-  2 e. RR
2 2lt8 
 |-  2 < 8
3 1 2 gtneii 
 |-  8 =/= 2
4 ipndx 
 |-  ( .i ` ndx ) = 8
5 plusgndx 
 |-  ( +g ` ndx ) = 2
6 4 5 neeq12i 
 |-  ( ( .i ` ndx ) =/= ( +g ` ndx ) <-> 8 =/= 2 )
7 3 6 mpbir 
 |-  ( .i ` ndx ) =/= ( +g ` ndx )