Theorem mapdpglem14

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	$$⊢\phi \to X\ne Q$$
	mapdpglem12.yn	$$⊢\phi \to Y\ne Q$$
	mapdpglem12.g0	$$⊢\phi \to z=0_C$$
Assertion	mapdpglem14	$$⊢\phi \to Y\in N\left(\left\{X\right\}\right)$$

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		mapdpglem12.yn	$$⊢\phi \to Y\ne Q$$
30		mapdpglem12.g0	$$⊢\phi \to z=0_C$$
31	1 3 8	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
32		eqid	$$⊢{+}_U={+}_U$$
33	4 32 5	lmodvnpcan	$$⊢\left(U\in \mathrm{LMod}\wedge Y\in V\wedge X\in V\right)\to \left(Y\underset{\dot{}}{-}X\right){+}_{U}X=Y$$
34	31 10 9 33	syl3anc	$$⊢\phi \to \left(Y\underset{\dot{}}{-}X\right){+}_{U}X=Y$$
35		eqid	$$⊢\mathrm{LSubSp}\left(U\right)=\mathrm{LSubSp}\left(U\right)$$
36	4 35 6	lspsncl	$$⊢\left(U\in \mathrm{LMod}\wedge X\in V\right)\to N\left(\left\{X\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
37	31 9 36	syl2anc	$$⊢\phi \to N\left(\left\{X\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
38		lmodgrp	$$⊢U\in \mathrm{LMod}\to U\in \mathrm{Grp}$$
39	31 38	syl	$$⊢\phi \to U\in \mathrm{Grp}$$
40		eqid	$$⊢{inv}_g\left(U\right)={inv}_g\left(U\right)$$
41	4 5 40	grpinvsub	$$⊢\left(U\in \mathrm{Grp}\wedge X\in V\wedge Y\in V\right)\to {inv}_g\left(U\right)\left(X\underset{\dot{}}{-}Y\right)=Y\underset{\dot{}}{-}X$$
42	39 9 10 41	syl3anc	$$⊢\phi \to {inv}_g\left(U\right)\left(X\underset{\dot{}}{-}Y\right)=Y\underset{\dot{}}{-}X$$
43	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 30	mapdpglem13	$$⊢\phi \to N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)\subseteq N\left(\left\{X\right\}\right)$$
44	4 5	lmodvsubcl	$$⊢\left(U\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{-}Y\in V$$
45	31 9 10 44	syl3anc	$$⊢\phi \to X\underset{\dot{}}{-}Y\in V$$
46	4 6	lspsnid	$$⊢\left(U\in \mathrm{LMod}\wedge X\underset{\dot{}}{-}Y\in V\right)\to X\underset{\dot{}}{-}Y\in N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)$$
47	31 45 46	syl2anc	$$⊢\phi \to X\underset{\dot{}}{-}Y\in N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)$$
48	43 47	sseldd	$$⊢\phi \to X\underset{\dot{}}{-}Y\in N\left(\left\{X\right\}\right)$$
49	35 40	lssvnegcl	$$⊢\left(U\in \mathrm{LMod}\wedge N\left(\left\{X\right\}\right)\in \mathrm{LSubSp}\left(U\right)\wedge X\underset{\dot{}}{-}Y\in N\left(\left\{X\right\}\right)\right)\to {inv}_g\left(U\right)\left(X\underset{\dot{}}{-}Y\right)\in N\left(\left\{X\right\}\right)$$
50	31 37 48 49	syl3anc	$$⊢\phi \to {inv}_g\left(U\right)\left(X\underset{\dot{}}{-}Y\right)\in N\left(\left\{X\right\}\right)$$
51	42 50	eqeltrrd	$$⊢\phi \to Y\underset{\dot{}}{-}X\in N\left(\left\{X\right\}\right)$$
52	4 6	lspsnid	$$⊢\left(U\in \mathrm{LMod}\wedge X\in V\right)\to X\in N\left(\left\{X\right\}\right)$$
53	31 9 52	syl2anc	$$⊢\phi \to X\in N\left(\left\{X\right\}\right)$$
54	32 35	lssvacl	$$⊢\left(\left(U\in \mathrm{LMod}\wedge N\left(\left\{X\right\}\right)\in \mathrm{LSubSp}\left(U\right)\right)\wedge \left(Y\underset{\dot{}}{-}X\in N\left(\left\{X\right\}\right)\wedge X\in N\left(\left\{X\right\}\right)\right)\right)\to \left(Y\underset{\dot{}}{-}X\right){+}_{U}X\in N\left(\left\{X\right\}\right)$$
55	31 37 51 53 54	syl22anc	$$⊢\phi \to \left(Y\underset{\dot{}}{-}X\right){+}_{U}X\in N\left(\left\{X\right\}\right)$$
56	34 55	eqeltrrd	$$⊢\phi \to Y\in N\left(\left\{X\right\}\right)$$