Theorem 00ply1bas

Description: Lemma for ply1basfvi and deg1fvi . (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Assertion	00ply1bas	$$⊢\varnothing ={\mathrm{Base}}_{{\mathrm{Poly}}_1\left(\varnothing \right)}$$

Proof

Step	Hyp	Ref	Expression
1		noel	$$⊢\neg a\in \varnothing$$
2		noel	$$⊢\neg a\left(1_{𝑜}\times \left\{0\right\}\right)\in \varnothing$$
3		eqid	$$⊢{\mathrm{Poly}}_1\left(\varnothing \right)={\mathrm{Poly}}_1\left(\varnothing \right)$$
4		eqid	$$⊢{\mathrm{Base}}_{{\mathrm{Poly}}_1\left(\varnothing \right)}={\mathrm{Base}}_{{\mathrm{Poly}}_1\left(\varnothing \right)}$$
5		base0	$$⊢\varnothing ={\mathrm{Base}}_{\varnothing }$$
6	3 4 5	ply1basf	$$⊢a\in {\mathrm{Base}}_{{\mathrm{Poly}}_1\left(\varnothing \right)}\to a:{{\mathbb{N}}_0}^{1_{𝑜}}⟶\varnothing$$
7		0nn0	$$⊢0\in {\mathbb{N}}_0$$
8	7	fconst6	$$⊢\left(1_{𝑜}\times \left\{0\right\}\right):1_{𝑜}⟶{\mathbb{N}}_0$$
9		nn0ex	$$⊢{\mathbb{N}}_0\in \mathrm{V}$$
10		1oex	$$⊢1_{𝑜}\in \mathrm{V}$$
11	9 10	elmap	$$⊢1_{𝑜}\times \left\{0\right\}\in \left({{\mathbb{N}}_0}^{1_{𝑜}}\right)\leftrightarrow \left(1_{𝑜}\times \left\{0\right\}\right):1_{𝑜}⟶{\mathbb{N}}_0$$
12	8 11	mpbir	$$⊢1_{𝑜}\times \left\{0\right\}\in \left({{\mathbb{N}}_0}^{1_{𝑜}}\right)$$
13		ffvelcdm	$$⊢\left(a:{{\mathbb{N}}_0}^{1_{𝑜}}⟶\varnothing \wedge 1_{𝑜}\times \left\{0\right\}\in \left({{\mathbb{N}}_0}^{1_{𝑜}}\right)\right)\to a\left(1_{𝑜}\times \left\{0\right\}\right)\in \varnothing$$
14	6 12 13	sylancl	$$⊢a\in {\mathrm{Base}}_{{\mathrm{Poly}}_1\left(\varnothing \right)}\to a\left(1_{𝑜}\times \left\{0\right\}\right)\in \varnothing$$
15	2 14	mto	$$⊢\neg a\in {\mathrm{Base}}_{{\mathrm{Poly}}_1\left(\varnothing \right)}$$
16	1 15	2false	$$⊢a\in \varnothing \leftrightarrow a\in {\mathrm{Base}}_{{\mathrm{Poly}}_1\left(\varnothing \right)}$$
17	16	eqriv	$$⊢\varnothing ={\mathrm{Base}}_{{\mathrm{Poly}}_1\left(\varnothing \right)}$$