Theorem lcfrlem37

Description: Lemma for lcfr . (Contributed by NM, 8-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	$$⊢C=J\left(X\right)\underset{\dot{}}{-}\left(\left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\right)$$
	lcfrlem37.g	$$⊢\phi \to G\in \mathrm{LSubSp}\left(D\right)$$
	lcfrlem37.gs	$$⊢\phi \to G\subseteq \left\{f\in \mathrm{LFnl}\left(U\right)|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}$$
	lcfrlem37.e	$$⊢E=\bigcup _{g\in G}\underset{\dot{}}{\perp }\left(L\left(g\right)\right)$$
	lcfrlem37.xe	$$⊢\phi \to X\in E$$
	lcfrlem37.ye	$$⊢\phi \to Y\in E$$
Assertion	lcfrlem37	$$⊢\phi \to X\underset{\dot{}}{+}Y\in E$$

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		lcfrlem37.g	$$⊢\phi \to G\in \mathrm{LSubSp}\left(D\right)$$
27		lcfrlem37.gs	$$⊢\phi \to G\subseteq \left\{f\in \mathrm{LFnl}\left(U\right)|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}$$
28		lcfrlem37.e	$$⊢E=\bigcup _{g\in G}\underset{\dot{}}{\perp }\left(L\left(g\right)\right)$$
29		lcfrlem37.xe	$$⊢\phi \to X\in E$$
30		lcfrlem37.ye	$$⊢\phi \to Y\in E$$
31		eqid	$$⊢\mathrm{LSubSp}\left(D\right)=\mathrm{LSubSp}\left(D\right)$$
32	1 3 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
33		eqid	$$⊢\mathrm{LFnl}\left(U\right)=\mathrm{LFnl}\left(U\right)$$
34		eqid	$$⊢0_D=0_D$$
35		eqid	$$⊢\left\{f\in \mathrm{LFnl}\left(U\right)|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}=\left\{f\in \mathrm{LFnl}\left(U\right)|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}$$
36		eldifsni	$$⊢X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)\to X\ne \underset{\dot{}}{0}$$
37	10 36	syl	$$⊢\phi \to X\ne \underset{\dot{}}{0}$$
38		eldifsn	$$⊢X\in \left(E\setminus \left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(X\in E\wedge X\ne \underset{\dot{}}{0}\right)$$
39	29 37 38	sylanbrc	$$⊢\phi \to X\in \left(E\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
40	1 2 3 4 5 14 15 17 6 33 20 21 34 35 18 9 31 26 27 28 39	lcfrlem16	$$⊢\phi \to J\left(X\right)\in G$$
41		eqid	$$⊢{\cdot }_D={\cdot }_D$$
42	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23	lcfrlem29	$$⊢\phi \to F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\in R$$
43		eldifsni	$$⊢Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)\to Y\ne \underset{\dot{}}{0}$$
44	11 43	syl	$$⊢\phi \to Y\ne \underset{\dot{}}{0}$$
45		eldifsn	$$⊢Y\in \left(E\setminus \left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(Y\in E\wedge Y\ne \underset{\dot{}}{0}\right)$$
46	30 44 45	sylanbrc	$$⊢\phi \to Y\in \left(E\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
47	1 2 3 4 5 14 15 17 6 33 20 21 34 35 18 9 31 26 27 28 46	lcfrlem16	$$⊢\phi \to J\left(Y\right)\in G$$
48	15 17 21 41 31 32 26 42 47	ldualssvscl	$$⊢\phi \to \left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\in G$$
49	21 24 31 32 26 40 48	ldualssvsubcl	$$⊢\phi \to J\left(X\right)\underset{\dot{}}{-}\left(\left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\right)\in G$$
50	25 49	eqeltrid	$$⊢\phi \to C\in G$$
51	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	lcfrlem36	$$⊢\phi \to X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(C\right)\right)$$
52		2fveq3	$$⊢g=C\to \underset{\dot{}}{\perp }\left(L\left(g\right)\right)=\underset{\dot{}}{\perp }\left(L\left(C\right)\right)$$
53	52	eleq2d	$$⊢g=C\to \left(X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(g\right)\right)\leftrightarrow X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(C\right)\right)\right)$$
54	53	rspcev	$$⊢\left(C\in G\wedge X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(C\right)\right)\right)\to \exists g\in GX\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(g\right)\right)$$
55	50 51 54	syl2anc	$$⊢\phi \to \exists g\in GX\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(g\right)\right)$$
56		eliun	$$⊢X\underset{\dot{}}{+}Y\in \bigcup _{g\in G}\underset{\dot{}}{\perp }\left(L\left(g\right)\right)\leftrightarrow \exists g\in GX\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(L\left(g\right)\right)$$
57	55 56	sylibr	$$⊢\phi \to X\underset{\dot{}}{+}Y\in \bigcup _{g\in G}\underset{\dot{}}{\perp }\left(L\left(g\right)\right)$$
58	57 28	eleqtrrdi	$$⊢\phi \to X\underset{\dot{}}{+}Y\in E$$