Theorem mapdh6b0N

Description: Lemmma for mapdh6N . (Contributed by NM, 23-Apr-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mapdh.q	$$⊢Q=0_C$$
	mapdh.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
	mapdh.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	mapdh.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	mapdh.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	mapdh.v	$$⊢V={\mathrm{Base}}_U$$
	mapdh.s	$$⊢\underset{\dot{}}{-}={-}_U$$
	mapdhc.o	$$⊢\underset{\dot{}}{0}=0_U$$
	mapdh.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	mapdh.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	mapdh.d	$$⊢D={\mathrm{Base}}_C$$
	mapdh.r	$$⊢R={-}_C$$
	mapdh.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
	mapdh.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	mapdhc.f	$$⊢\phi \to F\in D$$
	mapdh.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{F\right\}\right)$$
	mapdhcl.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh.p	$$⊢\underset{\dot{}}{+}={+}_U$$
	mapdh.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
	mapdh6b0.y	$$⊢\phi \to Y\in V$$
	mapdh6b0.z	$$⊢\phi \to Z\in V$$
	mapdh6b0.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\cap N\left(\left\{Y,Z\right\}\right)=\left\{\underset{\dot{}}{0}\right\}$$
Assertion	mapdh6b0N	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		mapdh.q	$$⊢Q=0_C$$
2		mapdh.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
3		mapdh.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		mapdh.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
5		mapdh.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
6		mapdh.v	$$⊢V={\mathrm{Base}}_U$$
7		mapdh.s	$$⊢\underset{\dot{}}{-}={-}_U$$
8		mapdhc.o	$$⊢\underset{\dot{}}{0}=0_U$$
9		mapdh.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
10		mapdh.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
11		mapdh.d	$$⊢D={\mathrm{Base}}_C$$
12		mapdh.r	$$⊢R={-}_C$$
13		mapdh.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
14		mapdh.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
15		mapdhc.f	$$⊢\phi \to F\in D$$
16		mapdh.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{F\right\}\right)$$
17		mapdhcl.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
18		mapdh.p	$$⊢\underset{\dot{}}{+}={+}_U$$
19		mapdh.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
20		mapdh6b0.y	$$⊢\phi \to Y\in V$$
21		mapdh6b0.z	$$⊢\phi \to Z\in V$$
22		mapdh6b0.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\cap N\left(\left\{Y,Z\right\}\right)=\left\{\underset{\dot{}}{0}\right\}$$
23		eqid	$$⊢\mathrm{LSubSp}\left(U\right)=\mathrm{LSubSp}\left(U\right)$$
24	3 5 14	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
25	3 5 14	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
26	6 23 9 25 20 21	lspprcl	$$⊢\phi \to N\left(\left\{Y,Z\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
27	6 8 9 23 24 26 17	lspdisjb	$$⊢\phi \to \left(\neg X\in N\left(\left\{Y,Z\right\}\right)\leftrightarrow N\left(\left\{X\right\}\right)\cap N\left(\left\{Y,Z\right\}\right)=\left\{\underset{\dot{}}{0}\right\}\right)$$
28	22 27	mpbird	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$