Theorem hdmaprnlem4N

Description: Part of proof of part 12 in Baer p. 49 line 19. (T* =) (Ft)* = Gs. (Contributed by NM, 27-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hdmaprnlem1.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	hdmaprnlem1.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	hdmaprnlem1.v	$$⊢V={\mathrm{Base}}_U$$
	hdmaprnlem1.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	hdmaprnlem1.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	hdmaprnlem1.l	$$⊢L=\mathrm{LSpan}\left(C\right)$$
	hdmaprnlem1.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	hdmaprnlem1.s	$$⊢S=\mathrm{HDMap}\left(K\right)\left(W\right)$$
	hdmaprnlem1.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	hdmaprnlem1.se	$$⊢\phi \to s\in \left(D\setminus \left\{Q\right\}\right)$$
	hdmaprnlem1.ve	$$⊢\phi \to v\in V$$
	hdmaprnlem1.e	$$⊢\phi \to M\left(N\left(\left\{v\right\}\right)\right)=L\left(\left\{s\right\}\right)$$
	hdmaprnlem1.ue	$$⊢\phi \to u\in V$$
	hdmaprnlem1.un	$$⊢\phi \to \neg u\in N\left(\left\{v\right\}\right)$$
	hdmaprnlem1.d	$$⊢D={\mathrm{Base}}_C$$
	hdmaprnlem1.q	$$⊢Q=0_C$$
	hdmaprnlem1.o	$$⊢\underset{\dot{}}{0}=0_U$$
	hdmaprnlem1.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
	hdmaprnlem1.t2	$$⊢\phi \to t\in \left(N\left(\left\{v\right\}\right)\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
Assertion	hdmaprnlem4N	$$⊢\phi \to M\left(N\left(\left\{t\right\}\right)\right)=L\left(\left\{s\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		hdmaprnlem1.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		hdmaprnlem1.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		hdmaprnlem1.v	$$⊢V={\mathrm{Base}}_U$$
4		hdmaprnlem1.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
5		hdmaprnlem1.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
6		hdmaprnlem1.l	$$⊢L=\mathrm{LSpan}\left(C\right)$$
7		hdmaprnlem1.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
8		hdmaprnlem1.s	$$⊢S=\mathrm{HDMap}\left(K\right)\left(W\right)$$
9		hdmaprnlem1.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
10		hdmaprnlem1.se	$$⊢\phi \to s\in \left(D\setminus \left\{Q\right\}\right)$$
11		hdmaprnlem1.ve	$$⊢\phi \to v\in V$$
12		hdmaprnlem1.e	$$⊢\phi \to M\left(N\left(\left\{v\right\}\right)\right)=L\left(\left\{s\right\}\right)$$
13		hdmaprnlem1.ue	$$⊢\phi \to u\in V$$
14		hdmaprnlem1.un	$$⊢\phi \to \neg u\in N\left(\left\{v\right\}\right)$$
15		hdmaprnlem1.d	$$⊢D={\mathrm{Base}}_C$$
16		hdmaprnlem1.q	$$⊢Q=0_C$$
17		hdmaprnlem1.o	$$⊢\underset{\dot{}}{0}=0_U$$
18		hdmaprnlem1.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
19		hdmaprnlem1.t2	$$⊢\phi \to t\in \left(N\left(\left\{v\right\}\right)\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
20		eqid	$$⊢\mathrm{LSubSp}\left(U\right)=\mathrm{LSubSp}\left(U\right)$$
21	1 2 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
22	3 20 4	lspsncl	$$⊢\left(U\in \mathrm{LMod}\wedge v\in V\right)\to N\left(\left\{v\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
23	21 11 22	syl2anc	$$⊢\phi \to N\left(\left\{v\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
24	19	eldifad	$$⊢\phi \to t\in N\left(\left\{v\right\}\right)$$
25	20 4 21 23 24	lspsnel5a	$$⊢\phi \to N\left(\left\{t\right\}\right)\subseteq N\left(\left\{v\right\}\right)$$
26	1 2 9	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
27	3 20	lss1	$$⊢U\in \mathrm{LMod}\to V\in \mathrm{LSubSp}\left(U\right)$$
28	21 27	syl	$$⊢\phi \to V\in \mathrm{LSubSp}\left(U\right)$$
29	20 4 21 28 11	lspsnel5a	$$⊢\phi \to N\left(\left\{v\right\}\right)\subseteq V$$
30	29	ssdifd	$$⊢\phi \to N\left(\left\{v\right\}\right)\setminus \left\{\underset{\dot{}}{0}\right\}\subseteq V\setminus \left\{\underset{\dot{}}{0}\right\}$$
31	30 19	sseldd	$$⊢\phi \to t\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
32	3 17 4 26 31 11	lspsncmp	$$⊢\phi \to \left(N\left(\left\{t\right\}\right)\subseteq N\left(\left\{v\right\}\right)\leftrightarrow N\left(\left\{t\right\}\right)=N\left(\left\{v\right\}\right)\right)$$
33	25 32	mpbid	$$⊢\phi \to N\left(\left\{t\right\}\right)=N\left(\left\{v\right\}\right)$$
34	33	fveq2d	$$⊢\phi \to M\left(N\left(\left\{t\right\}\right)\right)=M\left(N\left(\left\{v\right\}\right)\right)$$
35	34 12	eqtrd	$$⊢\phi \to M\left(N\left(\left\{t\right\}\right)\right)=L\left(\left\{s\right\}\right)$$