Theorem lcfrlem36

Description: Lemma for lcfr . (Contributed by NM, 6-Mar-2015)

	Ref	Expression
Hypotheses	lcfrlem17.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lcfrlem17.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
	lcfrlem17.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	lcfrlem17.v	$$⊢V={\mathrm{Base}}_U$$
	lcfrlem17.p	$$⊢\underset{\dot{}}{+}={+}_U$$
	lcfrlem17.z	$$⊢\underset{\dot{}}{0}=0_U$$
	lcfrlem17.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	lcfrlem17.a	$$⊢A=\mathrm{LSAtoms}\left(U\right)$$
	lcfrlem17.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	lcfrlem17.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	lcfrlem17.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	lcfrlem17.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
	lcfrlem22.b	$$⊢B=N\left(\left\{X,Y\right\}\right)\cap \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)$$
	lcfrlem24.t	$$⊢\underset{\dot{}}{·}={\cdot }_U$$
	lcfrlem24.s	$$⊢S=\mathrm{Scalar}\left(U\right)$$
	lcfrlem24.q	$$⊢Q=0_S$$
	lcfrlem24.r	$$⊢R={\mathrm{Base}}_S$$
	lcfrlem24.j	$$⊢J=\left(x\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)⟼\left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{x\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)\right)\right)\right)$$
	lcfrlem24.ib	$$⊢\phi \to I\in B$$
	lcfrlem24.l	$$⊢L=\mathrm{LKer}\left(U\right)$$
	lcfrlem25.d	$$⊢D=\mathrm{LDual}\left(U\right)$$
	lcfrlem28.jn	$$⊢\phi \to J\left(Y\right)\left(I\right)\ne Q$$
	lcfrlem29.i	$$⊢F={inv}_r\left(S\right)$$
	lcfrlem30.m	$$⊢\underset{\dot{}}{-}={-}_D$$
	lcfrlem30.c	$$⊢C=J\left(X\right)\underset{\dot{}}{-}\left(\left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\right)$$
Assertion	lcfrlem36	$$⊢\phi \to X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(C\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		lcfrlem17.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		lcfrlem17.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
3		lcfrlem17.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
4		lcfrlem17.v	$$⊢V={\mathrm{Base}}_U$$
5		lcfrlem17.p	$$⊢\underset{\dot{}}{+}={+}_U$$
6		lcfrlem17.z	$$⊢\underset{\dot{}}{0}=0_U$$
7		lcfrlem17.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
8		lcfrlem17.a	$$⊢A=\mathrm{LSAtoms}\left(U\right)$$
9		lcfrlem17.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
10		lcfrlem17.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
11		lcfrlem17.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
12		lcfrlem17.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
13		lcfrlem22.b	$$⊢B=N\left(\left\{X,Y\right\}\right)\cap \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)$$
14		lcfrlem24.t	$$⊢\underset{\dot{}}{·}={\cdot }_U$$
15		lcfrlem24.s	$$⊢S=\mathrm{Scalar}\left(U\right)$$
16		lcfrlem24.q	$$⊢Q=0_S$$
17		lcfrlem24.r	$$⊢R={\mathrm{Base}}_S$$
18		lcfrlem24.j	$$⊢J=\left(x\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)⟼\left(v\in V⟼\left(\iota k\in R|\exists w\in \underset{\dot{}}{\perp }\left(\left\{x\right\}\right)v=w\underset{\dot{}}{+}\left(k\underset{\dot{}}{·}x\right)\right)\right)\right)$$
19		lcfrlem24.ib	$$⊢\phi \to I\in B$$
20		lcfrlem24.l	$$⊢L=\mathrm{LKer}\left(U\right)$$
21		lcfrlem25.d	$$⊢D=\mathrm{LDual}\left(U\right)$$
22		lcfrlem28.jn	$$⊢\phi \to J\left(Y\right)\left(I\right)\ne Q$$
23		lcfrlem29.i	$$⊢F={inv}_r\left(S\right)$$
24		lcfrlem30.m	$$⊢\underset{\dot{}}{-}={-}_D$$
25		lcfrlem30.c	$$⊢C=J\left(X\right)\underset{\dot{}}{-}\left(\left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\right)$$
26	1 2 3 4 5 6 7 8 9 10 11 12	lcfrlem17	$$⊢\phi \to X\underset{\dot{}}{+}Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
27	26	eldifad	$$⊢\phi \to X\underset{\dot{}}{+}Y\in V$$
28	1 3 2 4 7 9 27	dochocsn	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)=N\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)$$
29	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25	lcfrlem35	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)=L\left(C\right)$$
30	29	fveq2d	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)=\underset{\dot{}}{\perp }\left(L\left(C\right)\right)$$
31	28 30	eqtr3d	$$⊢\phi \to N\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)=\underset{\dot{}}{\perp }\left(L\left(C\right)\right)$$
32		eqimss	$$⊢N\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)=\underset{\dot{}}{\perp }\left(L\left(C\right)\right)\to N\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\subseteq \underset{\dot{}}{\perp }\left(L\left(C\right)\right)$$
33	31 32	syl	$$⊢\phi \to N\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\subseteq \underset{\dot{}}{\perp }\left(L\left(C\right)\right)$$
34		eqid	$$⊢\mathrm{LSubSp}\left(U\right)=\mathrm{LSubSp}\left(U\right)$$
35	1 3 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
36		eqid	$$⊢\mathrm{LFnl}\left(U\right)=\mathrm{LFnl}\left(U\right)$$
37	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25	lcfrlem30	$$⊢\phi \to C\in \mathrm{LFnl}\left(U\right)$$
38	4 36 20 35 37	lkrssv	$$⊢\phi \to L\left(C\right)\subseteq V$$
39	1 3 4 34 2	dochlss	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge L\left(C\right)\subseteq V\right)\to \underset{\dot{}}{\perp }\left(L\left(C\right)\right)\in \mathrm{LSubSp}\left(U\right)$$
40	9 38 39	syl2anc	$$⊢\phi \to \underset{\dot{}}{\perp }\left(L\left(C\right)\right)\in \mathrm{LSubSp}\left(U\right)$$
41	4 34 7 35 40 27	ellspsn5b	$$⊢\phi \to \left(X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(C\right)\right)\leftrightarrow N\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\subseteq \underset{\dot{}}{\perp }\left(L\left(C\right)\right)\right)$$
42	33 41	mpbird	$$⊢\phi \to X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(C\right)\right)$$