Theorem hdmap1l6b

Description: Lemmma for hdmap1l6 . (Contributed by NM, 24-Apr-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	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=L\left(\left\{F\right\}\right)$$
	hdmap1l6b.y	$$⊢\phi \to Y=\underset{\dot{}}{0}$$
	hdmap1l6b.z	$$⊢\phi \to Z\in V$$
	hdmap1l6b.ne	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$
Assertion	hdmap1l6b	$$⊢\phi \to I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$

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		hdmap1l6b.y	$$⊢\phi \to Y=\underset{\dot{}}{0}$$
21		hdmap1l6b.z	$$⊢\phi \to Z\in V$$
22		hdmap1l6b.ne	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$
23	1 8 16	lcdlmod	$$⊢\phi \to C\in \mathrm{LMod}$$
24		lmodgrp	$$⊢C\in \mathrm{LMod}\to C\in \mathrm{Grp}$$
25	23 24	syl	$$⊢\phi \to C\in \mathrm{Grp}$$
26	1 2 16	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
27	18	eldifad	$$⊢\phi \to X\in V$$
28	1 2 16	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
29	3 6	lmod0vcl	$$⊢U\in \mathrm{LMod}\to \underset{\dot{}}{0}\in V$$
30	28 29	syl	$$⊢\phi \to \underset{\dot{}}{0}\in V$$
31	20 30	eqeltrd	$$⊢\phi \to Y\in V$$
32	3 7 26 27 31 21 22	lspindpi	$$⊢\phi \to \left(N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)\wedge N\left(\left\{X\right\}\right)\ne N\left(\left\{Z\right\}\right)\right)$$
33	32	simprd	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Z\right\}\right)$$
34	1 2 3 6 7 8 9 13 14 15 16 17 19 33 18 21	hdmap1cl	$$⊢\phi \to I\left(⟨X,F,Z⟩\right)\in D$$
35	9 10 12	grplid	$$⊢\left(C\in \mathrm{Grp}\wedge I\left(⟨X,F,Z⟩\right)\in D\right)\to Q\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)=I\left(⟨X,F,Z⟩\right)$$
36	25 34 35	syl2anc	$$⊢\phi \to Q\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)=I\left(⟨X,F,Z⟩\right)$$
37	20	oteq3d	$$⊢\phi \to ⟨X,F,Y⟩=⟨X,F,\underset{\dot{}}{0}⟩$$
38	37	fveq2d	$$⊢\phi \to I\left(⟨X,F,Y⟩\right)=I\left(⟨X,F,\underset{\dot{}}{0}⟩\right)$$
39	1 2 3 6 8 9 12 15 16 17 27	hdmap1val0	$$⊢\phi \to I\left(⟨X,F,\underset{\dot{}}{0}⟩\right)=Q$$
40	38 39	eqtrd	$$⊢\phi \to I\left(⟨X,F,Y⟩\right)=Q$$
41	40	oveq1d	$$⊢\phi \to I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)=Q\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$
42	20	oveq1d	$$⊢\phi \to Y\underset{\dot{}}{+}Z=\underset{\dot{}}{0}\underset{\dot{}}{+}Z$$
43		lmodgrp	$$⊢U\in \mathrm{LMod}\to U\in \mathrm{Grp}$$
44	28 43	syl	$$⊢\phi \to U\in \mathrm{Grp}$$
45	3 4 6	grplid	$$⊢\left(U\in \mathrm{Grp}\wedge Z\in V\right)\to \underset{\dot{}}{0}\underset{\dot{}}{+}Z=Z$$
46	44 21 45	syl2anc	$$⊢\phi \to \underset{\dot{}}{0}\underset{\dot{}}{+}Z=Z$$
47	42 46	eqtrd	$$⊢\phi \to Y\underset{\dot{}}{+}Z=Z$$
48	47	oteq3d	$$⊢\phi \to ⟨X,F,Y\underset{\dot{}}{+}Z⟩=⟨X,F,Z⟩$$
49	48	fveq2d	$$⊢\phi \to I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Z⟩\right)$$
50	36 41 49	3eqtr4rd	$$⊢\phi \to I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$