Theorem mapdpglem9

Description: Lemma for mapdpg . Baer p. 45, line 4: "...so that x would consequently belong to Fy." (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	$$⊢\phi \to X\ne Q$$
	mapdpglem4.g0	$$⊢\phi \to g=\underset{\dot{}}{0}$$
Assertion	mapdpglem9	$$⊢\phi \to X\in N\left(\left\{Y\right\}\right)$$

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		mapdpglem4.g0	$$⊢\phi \to g=\underset{\dot{}}{0}$$
30	1 3 8	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
31		eqid	$$⊢{+}_U={+}_U$$
32	4 31 5	lmodvnpcan	$$⊢\left(U\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to \left(X\underset{\dot{}}{-}Y\right){+}_{U}Y=X$$
33	30 9 10 32	syl3anc	$$⊢\phi \to \left(X\underset{\dot{}}{-}Y\right){+}_{U}Y=X$$
34		eqid	$$⊢\mathrm{LSubSp}\left(U\right)=\mathrm{LSubSp}\left(U\right)$$
35	4 34 6	lspsncl	$$⊢\left(U\in \mathrm{LMod}\wedge Y\in V\right)\to N\left(\left\{Y\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
36	30 10 35	syl2anc	$$⊢\phi \to N\left(\left\{Y\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
37	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29	mapdpglem8	$$⊢\phi \to N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\subseteq N\left(\left\{Y\right\}\right)$$
38	4 5	lmodvsubcl	$$⊢\left(U\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{-}Y\in V$$
39	30 9 10 38	syl3anc	$$⊢\phi \to X\underset{\dot{}}{-}Y\in V$$
40	4 6	lspsnid	$$⊢\left(U\in \mathrm{LMod}\wedge X\underset{\dot{}}{-}Y\in V\right)\to X\underset{\dot{}}{-}Y\in N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)$$
41	30 39 40	syl2anc	$$⊢\phi \to X\underset{\dot{}}{-}Y\in N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)$$
42	37 41	sseldd	$$⊢\phi \to X\underset{\dot{}}{-}Y\in N\left(\left\{Y\right\}\right)$$
43	4 6	lspsnid	$$⊢\left(U\in \mathrm{LMod}\wedge Y\in V\right)\to Y\in N\left(\left\{Y\right\}\right)$$
44	30 10 43	syl2anc	$$⊢\phi \to Y\in N\left(\left\{Y\right\}\right)$$
45	31 34	lssvacl	$$⊢\left(\left(U\in \mathrm{LMod}\wedge N\left(\left\{Y\right\}\right)\in \mathrm{LSubSp}\left(U\right)\right)\wedge \left(X\underset{\dot{}}{-}Y\in N\left(\left\{Y\right\}\right)\wedge Y\in N\left(\left\{Y\right\}\right)\right)\right)\to \left(X\underset{\dot{}}{-}Y\right){+}_{U}Y\in N\left(\left\{Y\right\}\right)$$
46	30 36 42 44 45	syl22anc	$$⊢\phi \to \left(X\underset{\dot{}}{-}Y\right){+}_{U}Y\in N\left(\left\{Y\right\}\right)$$
47	33 46	eqeltrrd	$$⊢\phi \to X\in N\left(\left\{Y\right\}\right)$$