Theorem mapdpglem16

Description: Lemma for mapdpg . Baer p. 45, line 7: "Likewise we see that z =/= 0." (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	$$⊢\phi \to X\ne Q$$
	mapdpglem12.yn	$$⊢\phi \to Y\ne Q$$
Assertion	mapdpglem16	$$⊢\phi \to z\ne 0_C$$

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		mapdpglem12.yn	$$⊢\phi \to Y\ne Q$$
30	8	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
31	9	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to X\in V$$
32	10	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to Y\in V$$
33	14	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to t\in \left(M\left(N\left(\left\{X\right\}\right)\right)\underset{\dot{}}{\oplus }M\left(N\left(\left\{Y\right\}\right)\right)\right)$$
34	19	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to G\in F$$
35	20	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{G\right\}\right)$$
36	22	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
37	23	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to M\left(N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\right)=J\left(\left\{t\right\}\right)$$
38	25	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to g\in B$$
39	26	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to z\in M\left(N\left(\left\{Y\right\}\right)\right)$$
40	27	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to t=\left(g\underset{\dot{}}{·}G\right)Rz$$
41	28	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to X\ne Q$$
42	29	adantr	$$⊢\left(\phi \wedge z=0_C\right)\to Y\ne Q$$
43		simpr	$$⊢\left(\phi \wedge z=0_C\right)\to z=0_C$$
44	1 2 3 4 5 6 7 30 31 32 11 12 13 33 15 16 17 18 34 35 21 36 37 24 38 39 40 41 42 43	mapdpglem15	$$⊢\left(\phi \wedge z=0_C\right)\to N\left(\left\{X\right\}\right)=N\left(\left\{Y\right\}\right)$$
45	44	ex	$$⊢\phi \to \left(z=0_C\to N\left(\left\{X\right\}\right)=N\left(\left\{Y\right\}\right)\right)$$
46	45	necon3d	$$⊢\phi \to \left(N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)\to z\ne 0_C\right)$$
47	22 46	mpd	$$⊢\phi \to z\ne 0_C$$