Metamath Proof Explorer


Theorem basendxnmulrndx

Description: The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020) (Proof shortened by AV, 28-Oct-2024)

Ref Expression
Assertion basendxnmulrndx ( Base ‘ ndx ) ≠ ( .r ‘ ndx )

Proof

Step Hyp Ref Expression
1 basendx ( Base ‘ ndx ) = 1
2 1re 1 ∈ ℝ
3 1lt3 1 < 3
4 2 3 ltneii 1 ≠ 3
5 mulrndx ( .r ‘ ndx ) = 3
6 4 5 neeqtrri 1 ≠ ( .r ‘ ndx )
7 1 6 eqnetri ( Base ‘ ndx ) ≠ ( .r ‘ ndx )