Theorem prjspnval2

Description: Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023)

	Ref	Expression
Hypotheses	prjspnval2.e	$$⊢\underset{\dot{}}{\sim }=\left\{⟨x,y⟩|\left(\left(x\in B\wedge y\in B\right)\wedge \exists l\in Sx=l\underset{\dot{}}{·}y\right)\right\}$$
	prjspnval2.w	$$⊢W=K\mathrm{freeLMod}\left(0\dots N\right)$$
	prjspnval2.b	$$⊢B={\mathrm{Base}}_W\setminus \left\{0_W\right\}$$
	prjspnval2.s	$$⊢S={\mathrm{Base}}_K$$
	prjspnval2.x	$$⊢\underset{\dot{}}{·}={\cdot }_W$$
Assertion	prjspnval2	$$⊢\left(N\in {\mathbb{N}}_0\wedge K\in \mathrm{DivRing}\right)\to N{\mathrm{\mathbb{P}𝕣𝕠𝕛}}_{n}K=B/\underset{\dot{}}{\sim }$$

Proof

Step	Hyp	Ref	Expression
1		prjspnval2.e	$$⊢\underset{\dot{}}{\sim }=\left\{⟨x,y⟩|\left(\left(x\in B\wedge y\in B\right)\wedge \exists l\in Sx=l\underset{\dot{}}{·}y\right)\right\}$$
2		prjspnval2.w	$$⊢W=K\mathrm{freeLMod}\left(0\dots N\right)$$
3		prjspnval2.b	$$⊢B={\mathrm{Base}}_W\setminus \left\{0_W\right\}$$
4		prjspnval2.s	$$⊢S={\mathrm{Base}}_K$$
5		prjspnval2.x	$$⊢\underset{\dot{}}{·}={\cdot }_W$$
6		prjspnval	$$⊢\left(N\in {\mathbb{N}}_0\wedge K\in \mathrm{DivRing}\right)\to N{\mathrm{\mathbb{P}𝕣𝕠𝕛}}_{n}K=\mathrm{\mathbb{P}𝕣𝕠𝕛}\left(K\mathrm{freeLMod}\left(0\dots N\right)\right)$$
7	2	fveq2i	$$⊢\mathrm{\mathbb{P}𝕣𝕠𝕛}\left(W\right)=\mathrm{\mathbb{P}𝕣𝕠𝕛}\left(K\mathrm{freeLMod}\left(0\dots N\right)\right)$$
8		ovex	$$⊢\left(0\dots N\right)\in \mathrm{V}$$
9	2	frlmlvec	$$⊢\left(K\in \mathrm{DivRing}\wedge \left(0\dots N\right)\in \mathrm{V}\right)\to W\in \mathrm{LVec}$$
10	8 9	mpan2	$$⊢K\in \mathrm{DivRing}\to W\in \mathrm{LVec}$$
11		eqid	$$⊢\mathrm{Scalar}\left(W\right)=\mathrm{Scalar}\left(W\right)$$
12		eqid	$$⊢{\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}={\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}$$
13	3 5 11 12	prjspval	$$⊢W\in \mathrm{LVec}\to \mathrm{\mathbb{P}𝕣𝕠𝕛}\left(W\right)=B/\left\{⟨x,y⟩|\left(\left(x\in B\wedge y\in B\right)\wedge \exists l\in {\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}x=l\underset{\dot{}}{·}y\right)\right\}$$
14	10 13	syl	$$⊢K\in \mathrm{DivRing}\to \mathrm{\mathbb{P}𝕣𝕠𝕛}\left(W\right)=B/\left\{⟨x,y⟩|\left(\left(x\in B\wedge y\in B\right)\wedge \exists l\in {\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}x=l\underset{\dot{}}{·}y\right)\right\}$$
15	1 2 3 4 5	prjspnerlem	$$⊢K\in \mathrm{DivRing}\to \underset{\dot{}}{\sim }=\left\{⟨x,y⟩|\left(\left(x\in B\wedge y\in B\right)\wedge \exists l\in {\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}x=l\underset{\dot{}}{·}y\right)\right\}$$
16	15	qseq2d	$$⊢K\in \mathrm{DivRing}\to B/\underset{\dot{}}{\sim }=B/\left\{⟨x,y⟩|\left(\left(x\in B\wedge y\in B\right)\wedge \exists l\in {\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}x=l\underset{\dot{}}{·}y\right)\right\}$$
17	14 16	eqtr4d	$$⊢K\in \mathrm{DivRing}\to \mathrm{\mathbb{P}𝕣𝕠𝕛}\left(W\right)=B/\underset{\dot{}}{\sim }$$
18	7 17	eqtr3id	$$⊢K\in \mathrm{DivRing}\to \mathrm{\mathbb{P}𝕣𝕠𝕛}\left(K\mathrm{freeLMod}\left(0\dots N\right)\right)=B/\underset{\dot{}}{\sim }$$
19	18	adantl	$$⊢\left(N\in {\mathbb{N}}_0\wedge K\in \mathrm{DivRing}\right)\to \mathrm{\mathbb{P}𝕣𝕠𝕛}\left(K\mathrm{freeLMod}\left(0\dots N\right)\right)=B/\underset{\dot{}}{\sim }$$
20	6 19	eqtrd	$$⊢\left(N\in {\mathbb{N}}_0\wedge K\in \mathrm{DivRing}\right)\to N{\mathrm{\mathbb{P}𝕣𝕠𝕛}}_{n}K=B/\underset{\dot{}}{\sim }$$