Theorem lcfrlem33

Description: Lemma for lcfr . (Contributed by NM, 10-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	$$⊢C=J\left(X\right)\underset{\dot{}}{-}\left(\left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\right)$$
	lcfrlem33.xi	$$⊢\phi \to J\left(X\right)\left(I\right)=Q$$
Assertion	lcfrlem33	$$⊢\phi \to C\ne 0_D$$

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		lcfrlem33.xi	$$⊢\phi \to J\left(X\right)\left(I\right)=Q$$
27	26	oveq2d	$$⊢\phi \to F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)=F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}Q$$
28	1 3 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
29	15	lmodring	$$⊢U\in \mathrm{LMod}\to S\in \mathrm{Ring}$$
30	28 29	syl	$$⊢\phi \to S\in \mathrm{Ring}$$
31	1 3 9	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
32	15	lvecdrng	$$⊢U\in \mathrm{LVec}\to S\in \mathrm{DivRing}$$
33	31 32	syl	$$⊢\phi \to S\in \mathrm{DivRing}$$
34		eqid	$$⊢\mathrm{LFnl}\left(U\right)=\mathrm{LFnl}\left(U\right)$$
35		eqid	$$⊢0_D=0_D$$
36		eqid	$$⊢\left\{f\in \mathrm{LFnl}\left(U\right)|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}=\left\{f\in \mathrm{LFnl}\left(U\right)|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}$$
37	1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 11	lcfrlem10	$$⊢\phi \to J\left(Y\right)\in \mathrm{LFnl}\left(U\right)$$
38	1 2 3 4 5 6 7 8 9 10 11 12 13	lcfrlem22	$$⊢\phi \to B\in A$$
39	4 8 28 38	lsatssv	$$⊢\phi \to B\subseteq V$$
40	39 19	sseldd	$$⊢\phi \to I\in V$$
41	15 17 4 34	lflcl	$$⊢\left(U\in \mathrm{LMod}\wedge J\left(Y\right)\in \mathrm{LFnl}\left(U\right)\wedge I\in V\right)\to J\left(Y\right)\left(I\right)\in R$$
42	28 37 40 41	syl3anc	$$⊢\phi \to J\left(Y\right)\left(I\right)\in R$$
43	17 16 23	drnginvrcl	$$⊢\left(S\in \mathrm{DivRing}\wedge J\left(Y\right)\left(I\right)\in R\wedge J\left(Y\right)\left(I\right)\ne Q\right)\to F\left(J\left(Y\right)\left(I\right)\right)\in R$$
44	33 42 22 43	syl3anc	$$⊢\phi \to F\left(J\left(Y\right)\left(I\right)\right)\in R$$
45		eqid	$$⊢{\cdot }_S={\cdot }_S$$
46	17 45 16	ringrz	$$⊢\left(S\in \mathrm{Ring}\wedge F\left(J\left(Y\right)\left(I\right)\right)\in R\right)\to F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}Q=Q$$
47	30 44 46	syl2anc	$$⊢\phi \to F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}Q=Q$$
48	27 47	eqtrd	$$⊢\phi \to F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)=Q$$
49	48	oveq1d	$$⊢\phi \to \left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)=Q{\cdot }_{D}J\left(Y\right)$$
50		eqid	$$⊢{\cdot }_D={\cdot }_D$$
51	34 15 16 21 50 35 28 37	ldual0vs	$$⊢\phi \to Q{\cdot }_{D}J\left(Y\right)=0_D$$
52	49 51	eqtrd	$$⊢\phi \to \left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)=0_D$$
53	52	oveq2d	$$⊢\phi \to J\left(X\right)\underset{\dot{}}{-}\left(\left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\right)=J\left(X\right)\underset{\dot{}}{-}{0}_D$$
54	21 28	ldualgrp	$$⊢\phi \to D\in \mathrm{Grp}$$
55		eqid	$$⊢{\mathrm{Base}}_D={\mathrm{Base}}_D$$
56	1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 10	lcfrlem10	$$⊢\phi \to J\left(X\right)\in \mathrm{LFnl}\left(U\right)$$
57	34 21 55 28 56	ldualelvbase	$$⊢\phi \to J\left(X\right)\in {\mathrm{Base}}_D$$
58	55 35 24	grpsubid1	$$⊢\left(D\in \mathrm{Grp}\wedge J\left(X\right)\in {\mathrm{Base}}_D\right)\to J\left(X\right)\underset{\dot{}}{-}{0}_D=J\left(X\right)$$
59	54 57 58	syl2anc	$$⊢\phi \to J\left(X\right)\underset{\dot{}}{-}{0}_D=J\left(X\right)$$
60	53 59	eqtrd	$$⊢\phi \to J\left(X\right)\underset{\dot{}}{-}\left(\left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\right)=J\left(X\right)$$
61	25 60	syl5eq	$$⊢\phi \to C=J\left(X\right)$$
62	1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 10	lcfrlem13	$$⊢\phi \to J\left(X\right)\in \left(\left\{f\in \mathrm{LFnl}\left(U\right)|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}\setminus \left\{0_D\right\}\right)$$
63		eldifsni	$$⊢J\left(X\right)\in \left(\left\{f\in \mathrm{LFnl}\left(U\right)|\underset{\dot{}}{\perp }\left(\underset{\dot{}}{\perp }\left(L\left(f\right)\right)\right)=L\left(f\right)\right\}\setminus \left\{0_D\right\}\right)\to J\left(X\right)\ne 0_D$$
64	62 63	syl	$$⊢\phi \to J\left(X\right)\ne 0_D$$
65	61 64	eqnetrd	$$⊢\phi \to C\ne 0_D$$