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)

	Ref	Expression
Assertion	basendxnmulrndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		df-base	$$⊢\mathrm{Base}=\mathrm{Slot}1$$
2		1nn	$$⊢1\in \mathbb{N}$$
3	1 2	ndxarg	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}=1$$
4		1re	$$⊢1\in ℝ$$
5		1lt3	$$⊢1<3$$
6	4 5	ltneii	$$⊢1\ne 3$$
7		mulrndx	$$⊢{\cdot }_{\mathrm{ndx}}=3$$
8	6 7	neeqtrri	$$⊢1\ne {\cdot }_{\mathrm{ndx}}$$
9	3 8	eqnetri	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$