Theorem hdmaprnlem3uN

Description: Part of proof of part 12 in Baer p. 49. (Contributed by NM, 29-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	$$⊢\underset{\dot{}}{✚}={+}_C$$
Assertion	hdmaprnlem3uN	$$⊢\phi \to N\left(\left\{u\right\}\right)\ne M^{-1}\left(L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\right)$$

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		eqid	$$⊢\mathrm{LSubSp}\left(U\right)=\mathrm{LSubSp}\left(U\right)$$
20	1 2 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
21	3 19 4	lspsncl	$$⊢\left(U\in \mathrm{LMod}\wedge u\in V\right)\to N\left(\left\{u\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
22	20 13 21	syl2anc	$$⊢\phi \to N\left(\left\{u\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
23	1 7 2 19 9 22	mapdcnvid1N	$$⊢\phi \to M^{-1}\left(M\left(N\left(\left\{u\right\}\right)\right)\right)=N\left(\left\{u\right\}\right)$$
24	1 2 3 4 5 6 7 8 9 13	hdmap10	$$⊢\phi \to M\left(N\left(\left\{u\right\}\right)\right)=L\left(\left\{S\left(u\right)\right\}\right)$$
25	1 5 9	lcdlvec	$$⊢\phi \to C\in \mathrm{LVec}$$
26	1 2 3 5 15 8 9 13	hdmapcl	$$⊢\phi \to S\left(u\right)\in D$$
27	1 2 3 4 5 6 7 8 9 10 11 12 13 14	hdmaprnlem1N	$$⊢\phi \to L\left(\left\{S\left(u\right)\right\}\right)\ne L\left(\left\{s\right\}\right)$$
28	15 18 16 6 25 26 10 27	lspindp3	$$⊢\phi \to L\left(\left\{S\left(u\right)\right\}\right)\ne L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$
29	24 28	eqnetrd	$$⊢\phi \to M\left(N\left(\left\{u\right\}\right)\right)\ne L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$
30	1 7 2 19 9 22	mapdcl	$$⊢\phi \to M\left(N\left(\left\{u\right\}\right)\right)\in \mathrm{ran}M$$
31	1 5 9	lcdlmod	$$⊢\phi \to C\in \mathrm{LMod}$$
32	10	eldifad	$$⊢\phi \to s\in D$$
33	15 18	lmodvacl	$$⊢\left(C\in \mathrm{LMod}\wedge S\left(u\right)\in D\wedge s\in D\right)\to S\left(u\right)\underset{\dot{}}{✚}s\in D$$
34	31 26 32 33	syl3anc	$$⊢\phi \to S\left(u\right)\underset{\dot{}}{✚}s\in D$$
35		eqid	$$⊢\mathrm{LSubSp}\left(C\right)=\mathrm{LSubSp}\left(C\right)$$
36	15 35 6	lspsncl	$$⊢\left(C\in \mathrm{LMod}\wedge S\left(u\right)\underset{\dot{}}{✚}s\in D\right)\to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\in \mathrm{LSubSp}\left(C\right)$$
37	31 34 36	syl2anc	$$⊢\phi \to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\in \mathrm{LSubSp}\left(C\right)$$
38	1 7 5 35 9	mapdrn2	$$⊢\phi \to \mathrm{ran}M=\mathrm{LSubSp}\left(C\right)$$
39	37 38	eleqtrrd	$$⊢\phi \to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\in \mathrm{ran}M$$
40	1 7 9 30 39	mapdcnv11N	$$⊢\phi \to \left(M^{-1}\left(M\left(N\left(\left\{u\right\}\right)\right)\right)=M^{-1}\left(L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\right)\leftrightarrow M\left(N\left(\left\{u\right\}\right)\right)=L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\right)$$
41	40	necon3bid	$$⊢\phi \to \left(M^{-1}\left(M\left(N\left(\left\{u\right\}\right)\right)\right)\ne M^{-1}\left(L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\right)\leftrightarrow M\left(N\left(\left\{u\right\}\right)\right)\ne L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\right)$$
42	29 41	mpbird	$$⊢\phi \to M^{-1}\left(M\left(N\left(\left\{u\right\}\right)\right)\right)\ne M^{-1}\left(L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\right)$$
43	23 42	eqnetrrd	$$⊢\phi \to N\left(\left\{u\right\}\right)\ne M^{-1}\left(L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\right)$$