Theorem lcfrlem34

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	$$⊢C=J\left(X\right)\underset{\dot{}}{-}\left(\left(F\left(J\left(Y\right)\left(I\right)\right){\cdot }_{S}J\left(X\right)\left(I\right)\right){\cdot }_{D}J\left(Y\right)\right)$$
Assertion	lcfrlem34	$$⊢\phi \to C\ne 0_D$$

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	9	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)=Q\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
27	10	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)=Q\right)\to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
28	11	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)=Q\right)\to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
29	12	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)=Q\right)\to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
30	19	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)=Q\right)\to I\in B$$
31	22	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)=Q\right)\to J\left(Y\right)\left(I\right)\ne Q$$
32		simpr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)=Q\right)\to J\left(X\right)\left(I\right)=Q$$
33	1 2 3 4 5 6 7 8 26 27 28 29 13 14 15 16 17 18 30 20 21 31 23 24 25 32	lcfrlem33	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)=Q\right)\to C\ne 0_D$$
34	9	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)\ne Q\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
35	10	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)\ne Q\right)\to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
36	11	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)\ne Q\right)\to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
37	12	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)\ne Q\right)\to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
38	19	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)\ne Q\right)\to I\in B$$
39	22	adantr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)\ne Q\right)\to J\left(Y\right)\left(I\right)\ne Q$$
40		simpr	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)\ne Q\right)\to J\left(X\right)\left(I\right)\ne Q$$
41	1 2 3 4 5 6 7 8 34 35 36 37 13 14 15 16 17 18 38 20 21 39 23 24 25 40	lcfrlem32	$$⊢\left(\phi \wedge J\left(X\right)\left(I\right)\ne Q\right)\to C\ne 0_D$$
42	33 41	pm2.61dane	$$⊢\phi \to C\ne 0_D$$