Theorem hdmaprnlem7N

Description: Part of proof of part 12 in Baer p. 49 line 19, s-St e. G(u'+s) = P*. (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	$$⊢\phi \to t\in \left(N\left(\left\{v\right\}\right)\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	hdmaprnlem1.p	$$⊢\underset{\dot{}}{+}={+}_U$$
	hdmaprnlem1.pt	$$⊢\phi \to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)=M\left(N\left(\left\{u\underset{\dot{}}{+}t\right\}\right)\right)$$
Assertion	hdmaprnlem7N	$$⊢\phi \to s{-}_{C}S\left(t\right)\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$

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		hdmaprnlem1.p	$$⊢\underset{\dot{}}{+}={+}_U$$
21		hdmaprnlem1.pt	$$⊢\phi \to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)=M\left(N\left(\left\{u\underset{\dot{}}{+}t\right\}\right)\right)$$
22		eqid	$$⊢{-}_C={-}_C$$
23	1 5 9	lcdlmod	$$⊢\phi \to C\in \mathrm{LMod}$$
24		lmodabl	$$⊢C\in \mathrm{LMod}\to C\in \mathrm{Abel}$$
25	23 24	syl	$$⊢\phi \to C\in \mathrm{Abel}$$
26	1 2 3 5 15 8 9 13	hdmapcl	$$⊢\phi \to S\left(u\right)\in D$$
27	10	eldifad	$$⊢\phi \to s\in D$$
28	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19	hdmaprnlem4tN	$$⊢\phi \to t\in V$$
29	1 2 3 5 15 8 9 28	hdmapcl	$$⊢\phi \to S\left(t\right)\in D$$
30	15 18 22 25 26 27 29 25 26 27 29	ablpnpcan	$$⊢\phi \to \left(S\left(u\right)\underset{\dot{}}{✚}s\right){-}_C\left(S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right)=s{-}_{C}S\left(t\right)$$
31	15 18	lmodvacl	$$⊢\left(C\in \mathrm{LMod}\wedge S\left(u\right)\in D\wedge s\in D\right)\to S\left(u\right)\underset{\dot{}}{✚}s\in D$$
32	23 26 27 31	syl3anc	$$⊢\phi \to S\left(u\right)\underset{\dot{}}{✚}s\in D$$
33		eqid	$$⊢\mathrm{LSubSp}\left(C\right)=\mathrm{LSubSp}\left(C\right)$$
34	15 33 6	lspsncl	$$⊢\left(C\in \mathrm{LMod}\wedge S\left(u\right)\underset{\dot{}}{✚}s\in D\right)\to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\in \mathrm{LSubSp}\left(C\right)$$
35	23 32 34	syl2anc	$$⊢\phi \to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\in \mathrm{LSubSp}\left(C\right)$$
36	15 6	lspsnid	$$⊢\left(C\in \mathrm{LMod}\wedge S\left(u\right)\underset{\dot{}}{✚}s\in D\right)\to S\left(u\right)\underset{\dot{}}{✚}s\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$
37	23 32 36	syl2anc	$$⊢\phi \to S\left(u\right)\underset{\dot{}}{✚}s\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$
38	15 18	lmodvacl	$$⊢\left(C\in \mathrm{LMod}\wedge S\left(u\right)\in D\wedge S\left(t\right)\in D\right)\to S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\in D$$
39	23 26 29 38	syl3anc	$$⊢\phi \to S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\in D$$
40	15 6	lspsnid	$$⊢\left(C\in \mathrm{LMod}\wedge S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\in D\right)\to S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right\}\right)$$
41	23 39 40	syl2anc	$$⊢\phi \to S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right\}\right)$$
42	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21	hdmaprnlem6N	$$⊢\phi \to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)=L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right\}\right)$$
43	41 42	eleqtrrd	$$⊢\phi \to S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$
44	22 33	lssvsubcl	$$⊢\left(\left(C\in \mathrm{LMod}\wedge L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\in \mathrm{LSubSp}\left(C\right)\right)\wedge \left(S\left(u\right)\underset{\dot{}}{✚}s\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\wedge S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)\right)\right)\to \left(S\left(u\right)\underset{\dot{}}{✚}s\right){-}_C\left(S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right)\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$
45	23 35 37 43 44	syl22anc	$$⊢\phi \to \left(S\left(u\right)\underset{\dot{}}{✚}s\right){-}_C\left(S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right)\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$
46	30 45	eqeltrrd	$$⊢\phi \to s{-}_{C}S\left(t\right)\in L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)$$