Theorem hdmaprnlem6N

Description: Part of proof of part 12 in Baer p. 49 line 18, G(u'+s) = G(u'+t). (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	hdmaprnlem6N	$$⊢\phi \to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)=L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right\}\right)$$

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	1 2 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
23	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19	hdmaprnlem4tN	$$⊢\phi \to t\in V$$
24	3 20	lmodvacl	$$⊢\left(U\in \mathrm{LMod}\wedge u\in V\wedge t\in V\right)\to u\underset{\dot{}}{+}t\in V$$
25	22 13 23 24	syl3anc	$$⊢\phi \to u\underset{\dot{}}{+}t\in V$$
26	1 2 3 4 5 6 7 8 9 25	hdmap10	$$⊢\phi \to M\left(N\left(\left\{u\underset{\dot{}}{+}t\right\}\right)\right)=L\left(\left\{S\left(u\underset{\dot{}}{+}t\right)\right\}\right)$$
27	1 2 3 20 5 18 8 9 13 23	hdmapadd	$$⊢\phi \to S\left(u\underset{\dot{}}{+}t\right)=S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)$$
28	27	sneqd	$$⊢\phi \to \left\{S\left(u\underset{\dot{}}{+}t\right)\right\}=\left\{S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right\}$$
29	28	fveq2d	$$⊢\phi \to L\left(\left\{S\left(u\underset{\dot{}}{+}t\right)\right\}\right)=L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right\}\right)$$
30	21 26 29	3eqtrd	$$⊢\phi \to L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}s\right\}\right)=L\left(\left\{S\left(u\right)\underset{\dot{}}{✚}S\left(t\right)\right\}\right)$$