Theorem plusgndxnmulrndx

Description: The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020)

	Ref	Expression
Assertion	plusgndxnmulrndx	⊢ ( +g ‘ ndx ) ≠ ( .r ‘ ndx )

Proof

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