Theorem hdmap1l6h

Description: Lemmma for hdmap1l6 . Part (6) of Baer p. 48 line 2. (Contributed by NM, 1-May-2015)

	Ref	Expression
Hypotheses	hdmap1l6.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	hdmap1l6.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	hdmap1l6.v	$$⊢V={\mathrm{Base}}_U$$
	hdmap1l6.p	$$⊢\underset{\dot{}}{+}={+}_U$$
	hdmap1l6.s	$$⊢\underset{\dot{}}{-}={-}_U$$
	hdmap1l6c.o	$$⊢\underset{\dot{}}{0}=0_U$$
	hdmap1l6.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	hdmap1l6.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	hdmap1l6.d	$$⊢D={\mathrm{Base}}_C$$
	hdmap1l6.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
	hdmap1l6.r	$$⊢R={-}_C$$
	hdmap1l6.q	$$⊢Q=0_C$$
	hdmap1l6.l	$$⊢L=\mathrm{LSpan}\left(C\right)$$
	hdmap1l6.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	hdmap1l6.i	$$⊢I=\mathrm{HDMap1}\left(K\right)\left(W\right)$$
	hdmap1l6.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	hdmap1l6.f	$$⊢\phi \to F\in D$$
	hdmap1l6cl.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	hdmap1l6.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=L\left(\left\{F\right\}\right)$$
	hdmap1l6d.xn	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$
	hdmap1l6d.yz	$$⊢\phi \to N\left(\left\{Y\right\}\right)=N\left(\left\{Z\right\}\right)$$
	hdmap1l6d.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	hdmap1l6d.z	$$⊢\phi \to Z\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	hdmap1l6d.w	$$⊢\phi \to w\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	hdmap1l6d.wn	$$⊢\phi \to \neg w\in N\left(\left\{X,Y\right\}\right)$$
Assertion	hdmap1l6h	$$⊢\phi \to I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$

Proof

Step	Hyp	Ref	Expression
1		hdmap1l6.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		hdmap1l6.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		hdmap1l6.v	$$⊢V={\mathrm{Base}}_U$$
4		hdmap1l6.p	$$⊢\underset{\dot{}}{+}={+}_U$$
5		hdmap1l6.s	$$⊢\underset{\dot{}}{-}={-}_U$$
6		hdmap1l6c.o	$$⊢\underset{\dot{}}{0}=0_U$$
7		hdmap1l6.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
8		hdmap1l6.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
9		hdmap1l6.d	$$⊢D={\mathrm{Base}}_C$$
10		hdmap1l6.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
11		hdmap1l6.r	$$⊢R={-}_C$$
12		hdmap1l6.q	$$⊢Q=0_C$$
13		hdmap1l6.l	$$⊢L=\mathrm{LSpan}\left(C\right)$$
14		hdmap1l6.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
15		hdmap1l6.i	$$⊢I=\mathrm{HDMap1}\left(K\right)\left(W\right)$$
16		hdmap1l6.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
17		hdmap1l6.f	$$⊢\phi \to F\in D$$
18		hdmap1l6cl.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
19		hdmap1l6.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=L\left(\left\{F\right\}\right)$$
20		hdmap1l6d.xn	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$
21		hdmap1l6d.yz	$$⊢\phi \to N\left(\left\{Y\right\}\right)=N\left(\left\{Z\right\}\right)$$
22		hdmap1l6d.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
23		hdmap1l6d.z	$$⊢\phi \to Z\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
24		hdmap1l6d.w	$$⊢\phi \to w\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
25		hdmap1l6d.wn	$$⊢\phi \to \neg w\in N\left(\left\{X,Y\right\}\right)$$
26	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25	hdmap1l6g	$$⊢\phi \to I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=\left(I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Y⟩\right)\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$
27	1 8 16	lcdlmod	$$⊢\phi \to C\in \mathrm{LMod}$$
28	1 2 16	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
29	24	eldifad	$$⊢\phi \to w\in V$$
30	18	eldifad	$$⊢\phi \to X\in V$$
31	22	eldifad	$$⊢\phi \to Y\in V$$
32	3 7 28 29 30 31 25	lspindpi	$$⊢\phi \to \left(N\left(\left\{w\right\}\right)\ne N\left(\left\{X\right\}\right)\wedge N\left(\left\{w\right\}\right)\ne N\left(\left\{Y\right\}\right)\right)$$
33	32	simpld	$$⊢\phi \to N\left(\left\{w\right\}\right)\ne N\left(\left\{X\right\}\right)$$
34	33	necomd	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{w\right\}\right)$$
35	1 2 3 6 7 8 9 13 14 15 16 17 19 34 18 29	hdmap1cl	$$⊢\phi \to I\left(⟨X,F,w⟩\right)\in D$$
36	23	eldifad	$$⊢\phi \to Z\in V$$
37	3 7 28 30 31 36 20	lspindpi	$$⊢\phi \to \left(N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)\wedge N\left(\left\{X\right\}\right)\ne N\left(\left\{Z\right\}\right)\right)$$
38	37	simpld	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
39	1 2 3 6 7 8 9 13 14 15 16 17 19 38 18 31	hdmap1cl	$$⊢\phi \to I\left(⟨X,F,Y⟩\right)\in D$$
40	37	simprd	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Z\right\}\right)$$
41	1 2 3 6 7 8 9 13 14 15 16 17 19 40 18 36	hdmap1cl	$$⊢\phi \to I\left(⟨X,F,Z⟩\right)\in D$$
42	9 10	lmodass	$$⊢\left(C\in \mathrm{LMod}\wedge \left(I\left(⟨X,F,w⟩\right)\in D\wedge I\left(⟨X,F,Y⟩\right)\in D\wedge I\left(⟨X,F,Z⟩\right)\in D\right)\right)\to \left(I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Y⟩\right)\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)=I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}\left(I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\right)$$
43	27 35 39 41 42	syl13anc	$$⊢\phi \to \left(I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Y⟩\right)\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)=I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}\left(I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\right)$$
44	26 43	eqtrd	$$⊢\phi \to I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}\left(I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\right)$$
45	3 4 6 7 28 18 22 23 24 21 38 25	mapdindp1	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\underset{\dot{}}{+}Z\right\}\right)$$
46	1 2 16	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
47	3 4	lmodvacl	$$⊢\left(U\in \mathrm{LMod}\wedge Y\in V\wedge Z\in V\right)\to Y\underset{\dot{}}{+}Z\in V$$
48	46 31 36 47	syl3anc	$$⊢\phi \to Y\underset{\dot{}}{+}Z\in V$$
49	1 2 3 6 7 8 9 13 14 15 16 17 19 45 18 48	hdmap1cl	$$⊢\phi \to I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)\in D$$
50	9 10	lmodvacl	$$⊢\left(C\in \mathrm{LMod}\wedge I\left(⟨X,F,Y⟩\right)\in D\wedge I\left(⟨X,F,Z⟩\right)\in D\right)\to I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\in D$$
51	27 39 41 50	syl3anc	$$⊢\phi \to I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\in D$$
52	9 10	lmodlcan	$$⊢\left(C\in \mathrm{LMod}\wedge \left(I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)\in D\wedge I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\in D\wedge I\left(⟨X,F,w⟩\right)\in D\right)\right)\to \left(I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}\left(I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\right)\leftrightarrow I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\right)$$
53	27 49 51 35 52	syl13anc	$$⊢\phi \to \left(I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,w⟩\right)\underset{\dot{}}{✚}\left(I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\right)\leftrightarrow I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)\right)$$
54	44 53	mpbid	$$⊢\phi \to I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$