Theorem mapdpglem12

Description: Lemma for mapdpg . TODO: Can some commonality with mapdpglem6 through mapdpglem11 be exploited? Also, some consolidation of small lemmas here could be done. (Contributed by NM, 18-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	mapdpglem12	$$⊢\phi \to t\in M\left(N\left(\left\{X\right\}\right)\right)$$

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 7 8	lcdlmod	$$⊢\phi \to C\in \mathrm{LMod}$$
32		eqid	$$⊢\mathrm{LSubSp}\left(U\right)=\mathrm{LSubSp}\left(U\right)$$
33		eqid	$$⊢\mathrm{LSubSp}\left(C\right)=\mathrm{LSubSp}\left(C\right)$$
34	1 3 8	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
35	4 32 6	lspsncl	$$⊢\left(U\in \mathrm{LMod}\wedge X\in V\right)\to N\left(\left\{X\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
36	34 9 35	syl2anc	$$⊢\phi \to N\left(\left\{X\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
37	1 2 3 32 7 33 8 36	mapdcl2	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)\in \mathrm{LSubSp}\left(C\right)$$
38	13 12	lspsnid	$$⊢\left(C\in \mathrm{LMod}\wedge G\in F\right)\to G\in J\left(\left\{G\right\}\right)$$
39	31 19 38	syl2anc	$$⊢\phi \to G\in J\left(\left\{G\right\}\right)$$
40	39 20	eleqtrrd	$$⊢\phi \to G\in M\left(N\left(\left\{X\right\}\right)\right)$$
41	1 3 15 16 7 13 17 33 8 37 25 40	lcdlssvscl	$$⊢\phi \to g\underset{\dot{}}{·}G\in M\left(N\left(\left\{X\right\}\right)\right)$$
42		eqid	$$⊢0_C=0_C$$
43	42 33	lss0cl	$$⊢\left(C\in \mathrm{LMod}\wedge M\left(N\left(\left\{X\right\}\right)\right)\in \mathrm{LSubSp}\left(C\right)\right)\to 0_C\in M\left(N\left(\left\{X\right\}\right)\right)$$
44	31 37 43	syl2anc	$$⊢\phi \to 0_C\in M\left(N\left(\left\{X\right\}\right)\right)$$
45	30 44	eqeltrd	$$⊢\phi \to z\in M\left(N\left(\left\{X\right\}\right)\right)$$
46	18 33	lssvsubcl	$$⊢\left(\left(C\in \mathrm{LMod}\wedge M\left(N\left(\left\{X\right\}\right)\right)\in \mathrm{LSubSp}\left(C\right)\right)\wedge \left(g\underset{\dot{}}{·}G\in M\left(N\left(\left\{X\right\}\right)\right)\wedge z\in M\left(N\left(\left\{X\right\}\right)\right)\right)\right)\to \left(g\underset{\dot{}}{·}G\right)Rz\in M\left(N\left(\left\{X\right\}\right)\right)$$
47	31 37 41 45 46	syl22anc	$$⊢\phi \to \left(g\underset{\dot{}}{·}G\right)Rz\in M\left(N\left(\left\{X\right\}\right)\right)$$
48	27 47	eqeltrd	$$⊢\phi \to t\in M\left(N\left(\left\{X\right\}\right)\right)$$