Theorem mapdpglem11

Description: Lemma for mapdpg . (Contributed by NM, 20-Mar-2015)

	Ref	Expression
Hypotheses	mapdpglem.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	mapdpglem.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	mapdpglem.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	mapdpglem.v	$$⊢V={\mathrm{Base}}_U$$
	mapdpglem.s	$$⊢\underset{\dot{}}{-}={-}_U$$
	mapdpglem.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	mapdpglem.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	mapdpglem.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	mapdpglem.x	$$⊢\phi \to X\in V$$
	mapdpglem.y	$$⊢\phi \to Y\in V$$
	mapdpglem1.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(C\right)$$
	mapdpglem2.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
	mapdpglem3.f	$$⊢F={\mathrm{Base}}_C$$
	mapdpglem3.te	$$⊢\phi \to t\in \left(M\left(N\left(\left\{X\right\}\right)\right)\underset{\dot{}}{\oplus }M\left(N\left(\left\{Y\right\}\right)\right)\right)$$
	mapdpglem3.a	$$⊢A=\mathrm{Scalar}\left(U\right)$$
	mapdpglem3.b	$$⊢B={\mathrm{Base}}_A$$
	mapdpglem3.t	$$⊢\underset{\dot{}}{·}={\cdot }_C$$
	mapdpglem3.r	$$⊢R={-}_C$$
	mapdpglem3.g	$$⊢\phi \to G\in F$$
	mapdpglem3.e	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{G\right\}\right)$$
	mapdpglem4.q	$$⊢Q=0_U$$
	mapdpglem.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
	mapdpglem4.jt	$$⊢\phi \to M\left(N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\right)=J\left(\left\{t\right\}\right)$$
	mapdpglem4.z	$$⊢\underset{\dot{}}{0}=0_A$$
	mapdpglem4.g4	$$⊢\phi \to g\in B$$
	mapdpglem4.z4	$$⊢\phi \to z\in M\left(N\left(\left\{Y\right\}\right)\right)$$
	mapdpglem4.t4	$$⊢\phi \to t=\left(g\underset{\dot{}}{·}G\right)Rz$$
	mapdpglem4.xn	$$⊢\phi \to X\ne Q$$
Assertion	mapdpglem11	$$⊢\phi \to g\ne \underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		mapdpglem.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		mapdpglem.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
3		mapdpglem.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
4		mapdpglem.v	$$⊢V={\mathrm{Base}}_U$$
5		mapdpglem.s	$$⊢\underset{\dot{}}{-}={-}_U$$
6		mapdpglem.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
7		mapdpglem.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
8		mapdpglem.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
9		mapdpglem.x	$$⊢\phi \to X\in V$$
10		mapdpglem.y	$$⊢\phi \to Y\in V$$
11		mapdpglem1.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(C\right)$$
12		mapdpglem2.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
13		mapdpglem3.f	$$⊢F={\mathrm{Base}}_C$$
14		mapdpglem3.te	$$⊢\phi \to t\in \left(M\left(N\left(\left\{X\right\}\right)\right)\underset{\dot{}}{\oplus }M\left(N\left(\left\{Y\right\}\right)\right)\right)$$
15		mapdpglem3.a	$$⊢A=\mathrm{Scalar}\left(U\right)$$
16		mapdpglem3.b	$$⊢B={\mathrm{Base}}_A$$
17		mapdpglem3.t	$$⊢\underset{\dot{}}{·}={\cdot }_C$$
18		mapdpglem3.r	$$⊢R={-}_C$$
19		mapdpglem3.g	$$⊢\phi \to G\in F$$
20		mapdpglem3.e	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{G\right\}\right)$$
21		mapdpglem4.q	$$⊢Q=0_U$$
22		mapdpglem.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
23		mapdpglem4.jt	$$⊢\phi \to M\left(N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\right)=J\left(\left\{t\right\}\right)$$
24		mapdpglem4.z	$$⊢\underset{\dot{}}{0}=0_A$$
25		mapdpglem4.g4	$$⊢\phi \to g\in B$$
26		mapdpglem4.z4	$$⊢\phi \to z\in M\left(N\left(\left\{Y\right\}\right)\right)$$
27		mapdpglem4.t4	$$⊢\phi \to t=\left(g\underset{\dot{}}{·}G\right)Rz$$
28		mapdpglem4.xn	$$⊢\phi \to X\ne Q$$
29	8	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
30	9	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to X\in V$$
31	10	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to Y\in V$$
32	14	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to t\in \left(M\left(N\left(\left\{X\right\}\right)\right)\underset{\dot{}}{\oplus }M\left(N\left(\left\{Y\right\}\right)\right)\right)$$
33	19	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to G\in F$$
34	20	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{G\right\}\right)$$
35	22	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
36	23	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to M\left(N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\right)=J\left(\left\{t\right\}\right)$$
37	25	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to g\in B$$
38	26	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to z\in M\left(N\left(\left\{Y\right\}\right)\right)$$
39	27	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to t=\left(g\underset{\dot{}}{·}G\right)Rz$$
40	28	adantr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to X\ne Q$$
41		simpr	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to g=\underset{\dot{}}{0}$$
42	1 2 3 4 5 6 7 29 30 31 11 12 13 32 15 16 17 18 33 34 21 35 36 24 37 38 39 40 41	mapdpglem10	$$⊢\left(\phi \wedge g=\underset{\dot{}}{0}\right)\to N\left(\left\{X\right\}\right)=N\left(\left\{Y\right\}\right)$$
43	42	ex	$$⊢\phi \to \left(g=\underset{\dot{}}{0}\to N\left(\left\{X\right\}\right)=N\left(\left\{Y\right\}\right)\right)$$
44	43	necon3d	$$⊢\phi \to \left(N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)\to g\ne \underset{\dot{}}{0}\right)$$
45	22 44	mpd	$$⊢\phi \to g\ne \underset{\dot{}}{0}$$