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

Proof

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