Theorem zlmodzxzldeplem2

Description: Lemma 2 for zlmodzxzldep . (Contributed by AV, 24-May-2019) (Revised by AV, 30-Jul-2019)

	Ref	Expression
Hypotheses	zlmodzxzldep.z	$$⊢Z={\mathbb{Z}}_{\mathrm{ring}}\mathrm{freeLMod}\left\{0,1\right\}$$
	zlmodzxzldep.a	$$⊢A=\left\{⟨0,3⟩,⟨1,6⟩\right\}$$
	zlmodzxzldep.b	$$⊢B=\left\{⟨0,2⟩,⟨1,4⟩\right\}$$
	zlmodzxzldeplem.f	$$⊢F=\left\{⟨A,2⟩,⟨B,-3⟩\right\}$$
Assertion	zlmodzxzldeplem2	$$⊢{finSupp}_0\left(F\right)$$

Proof

Step	Hyp	Ref	Expression
1		zlmodzxzldep.z	$$⊢Z={\mathbb{Z}}_{\mathrm{ring}}\mathrm{freeLMod}\left\{0,1\right\}$$
2		zlmodzxzldep.a	$$⊢A=\left\{⟨0,3⟩,⟨1,6⟩\right\}$$
3		zlmodzxzldep.b	$$⊢B=\left\{⟨0,2⟩,⟨1,4⟩\right\}$$
4		zlmodzxzldeplem.f	$$⊢F=\left\{⟨A,2⟩,⟨B,-3⟩\right\}$$
5	1 2 3 4	zlmodzxzldeplem1	$$⊢F\in \left({\mathbb{Z}}^{\left\{A,B\right\}}\right)$$
6		elmapi	$$⊢F\in \left({\mathbb{Z}}^{\left\{A,B\right\}}\right)\to F:\left\{A,B\right\}⟶\mathbb{Z}$$
7		prfi	$$⊢\left\{A,B\right\}\in \mathrm{Fin}$$
8	7	a1i	$$⊢F\in \left({\mathbb{Z}}^{\left\{A,B\right\}}\right)\to \left\{A,B\right\}\in \mathrm{Fin}$$
9		c0ex	$$⊢0\in \mathrm{V}$$
10	9	a1i	$$⊢F\in \left({\mathbb{Z}}^{\left\{A,B\right\}}\right)\to 0\in \mathrm{V}$$
11	6 8 10	fdmfifsupp	$$⊢F\in \left({\mathbb{Z}}^{\left\{A,B\right\}}\right)\to {finSupp}_0\left(F\right)$$
12	5 11	ax-mp	$$⊢{finSupp}_0\left(F\right)$$