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	$$⊢{+}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		plusgndx	$$⊢{+}_{\mathrm{ndx}}=2$$
2		2re	$$⊢2\in ℝ$$
3		2lt3	$$⊢2<3$$
4	2 3	ltneii	$$⊢2\ne 3$$
5		mulrndx	$$⊢{\cdot }_{\mathrm{ndx}}=3$$
6	4 5	neeqtrri	$$⊢2\ne {\cdot }_{\mathrm{ndx}}$$
7	1 6	eqnetri	$$⊢{+}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$