Theorem lcfrlem8

Description: Lemma for lcf1o and lcfr . (Contributed by NM, 21-Feb-2015)

	Ref	Expression
Hypotheses	lcf1o.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lcf1o.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
	lcf1o.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	lcf1o.v	$$⊢V={\mathrm{Base}}_U$$
	lcf1o.a	$$⊢\underset{\dot{}}{+}={+}_U$$
	lcf1o.t	$$⊢\underset{\dot{}}{·}={\cdot }_U$$
	lcf1o.s	$$⊢S=\mathrm{Scalar}\left(U\right)$$
	lcf1o.r	$$⊢R={\mathrm{Base}}_S$$
	lcf1o.z	$$⊢\underset{\dot{}}{0}=0_U$$
	lcf1o.f	$$⊢F=\mathrm{LFnl}\left(U\right)$$
	lcf1o.l	$$⊢L=\mathrm{LKer}\left(U\right)$$
	lcf1o.d	$$⊢D=\mathrm{LDual}\left(U\right)$$
	lcf1o.q	$$⊢Q=0_D$$
	lcf1o.c	$$⊢C=\left\{f\in F|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}$$
	lcf1o.j	$$⊢J=\left(x\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)⟼\left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{x\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)\right)\right)\right)$$
	lcflo.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	lcfrlem8.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
Assertion	lcfrlem8	$$⊢\phi \to J\left(X\right)=\left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{X\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}X\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		lcf1o.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		lcf1o.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
3		lcf1o.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
4		lcf1o.v	$$⊢V={\mathrm{Base}}_U$$
5		lcf1o.a	$$⊢\underset{\dot{}}{+}={+}_U$$
6		lcf1o.t	$$⊢\underset{\dot{}}{·}={\cdot }_U$$
7		lcf1o.s	$$⊢S=\mathrm{Scalar}\left(U\right)$$
8		lcf1o.r	$$⊢R={\mathrm{Base}}_S$$
9		lcf1o.z	$$⊢\underset{\dot{}}{0}=0_U$$
10		lcf1o.f	$$⊢F=\mathrm{LFnl}\left(U\right)$$
11		lcf1o.l	$$⊢L=\mathrm{LKer}\left(U\right)$$
12		lcf1o.d	$$⊢D=\mathrm{LDual}\left(U\right)$$
13		lcf1o.q	$$⊢Q=0_D$$
14		lcf1o.c	$$⊢C=\left\{f\in F|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}$$
15		lcf1o.j	$$⊢J=\left(x\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)⟼\left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{x\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)\right)\right)\right)$$
16		lcflo.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
17		lcfrlem8.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
18		sneq	$$⊢x=X\to \left\{x\right\}=\left\{X\right\}$$
19	18	fveq2d	$$⊢x=X\to \underset{\dot{}}{\perp }\left(\left\{x\right\}\right)=\underset{\dot{}}{\perp }\left(\left\{X\right\}\right)$$
20		oveq2	$$⊢x=X\to k\underset{\dot{}}{·}x=k\underset{\dot{}}{·}X$$
21	20	oveq2d	$$⊢x=X\to w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}X\right)$$
22	21	eqeq2d	$$⊢x=X\to \left(v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)\leftrightarrow v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}X\right)\right)$$
23	19 22	rexeqbidv	$$⊢x=X\to \left(\exists w\in \underset{\dot{}}{\perp }\left(\left\{x\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)\leftrightarrow \exists w\in \underset{\dot{}}{\perp }\left(\left\{X\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}X\right)\right)$$
24	23	riotabidv	$$⊢x=X\to \left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{x\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)\right)=\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{X\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}X\right)\right)$$
25	24	mpteq2dv	$$⊢x=X\to \left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{x\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)\right)\right)=\left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{X\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}X\right)\right)\right)$$
26	25 15 4	mptfvmpt	$$⊢X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)\to J\left(X\right)=\left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{X\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}X\right)\right)\right)$$
27	17 26	syl	$$⊢\phi \to J\left(X\right)=\left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{X\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}X\right)\right)\right)$$