Theorem baseltedgf

Description: The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020)

	Ref	Expression
Assertion	baseltedgf	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}<\mathrm{ef}\left(\mathrm{ndx}\right)$$

Proof

Step	Hyp	Ref	Expression
1		1nn	$$⊢1\in \mathbb{N}$$
2		8nn0	$$⊢8\in {\mathbb{N}}_0$$
3		1nn0	$$⊢1\in {\mathbb{N}}_0$$
4		1lt10	$$⊢1<10$$
5	1 2 3 4	declti	$$⊢1<18$$
6		df-base	$$⊢\mathrm{Base}=\mathrm{Slot}1$$
7	6 1	ndxarg	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}=1$$
8		df-edgf	$$⊢\mathrm{ef}=\mathrm{Slot}18$$
9		8nn	$$⊢8\in \mathbb{N}$$
10	3 9	decnncl	$$⊢18\in \mathbb{N}$$
11	8 10	ndxarg	$$⊢\mathrm{ef}\left(\mathrm{ndx}\right)=18$$
12	5 7 11	3brtr4i	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}<\mathrm{ef}\left(\mathrm{ndx}\right)$$