Theorem mapdh6eN

Description: Lemmma for mapdh6N . Part (6) in Baer p. 47 line 38. (Contributed by NM, 1-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mapdh.q	$$⊢Q=0_C$$
	mapdh.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
	mapdh.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	mapdh.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	mapdh.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	mapdh.v	$$⊢V={\mathrm{Base}}_U$$
	mapdh.s	$$⊢\underset{\dot{}}{-}={-}_U$$
	mapdhc.o	$$⊢\underset{\dot{}}{0}=0_U$$
	mapdh.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	mapdh.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	mapdh.d	$$⊢D={\mathrm{Base}}_C$$
	mapdh.r	$$⊢R={-}_C$$
	mapdh.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
	mapdh.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	mapdhc.f	$$⊢\phi \to F\in D$$
	mapdh.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{F\right\}\right)$$
	mapdhcl.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh.p	$$⊢\underset{\dot{}}{+}={+}_U$$
	mapdh.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
	mapdh6d.xn	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$
	mapdh6d.yz	$$⊢\phi \to N\left(\left\{Y\right\}\right)=N\left(\left\{Z\right\}\right)$$
	mapdh6d.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh6d.z	$$⊢\phi \to Z\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh6d.w	$$⊢\phi \to w\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh6d.wn	$$⊢\phi \to \neg w\in N\left(\left\{X,Y\right\}\right)$$
Assertion	mapdh6eN	$$⊢\phi \to I\left(⟨X,F,\left(w\underset{\dot{}}{+}Y\right)\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,w\underset{\dot{}}{+}Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$

Proof

Step	Hyp	Ref	Expression
1		mapdh.q	$$⊢Q=0_C$$
2		mapdh.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
3		mapdh.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		mapdh.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
5		mapdh.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
6		mapdh.v	$$⊢V={\mathrm{Base}}_U$$
7		mapdh.s	$$⊢\underset{\dot{}}{-}={-}_U$$
8		mapdhc.o	$$⊢\underset{\dot{}}{0}=0_U$$
9		mapdh.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
10		mapdh.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
11		mapdh.d	$$⊢D={\mathrm{Base}}_C$$
12		mapdh.r	$$⊢R={-}_C$$
13		mapdh.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
14		mapdh.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
15		mapdhc.f	$$⊢\phi \to F\in D$$
16		mapdh.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{F\right\}\right)$$
17		mapdhcl.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
18		mapdh.p	$$⊢\underset{\dot{}}{+}={+}_U$$
19		mapdh.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
20		mapdh6d.xn	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$
21		mapdh6d.yz	$$⊢\phi \to N\left(\left\{Y\right\}\right)=N\left(\left\{Z\right\}\right)$$
22		mapdh6d.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
23		mapdh6d.z	$$⊢\phi \to Z\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
24		mapdh6d.w	$$⊢\phi \to w\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
25		mapdh6d.wn	$$⊢\phi \to \neg w\in N\left(\left\{X,Y\right\}\right)$$
26	3 5 14	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
27	24	eldifad	$$⊢\phi \to w\in V$$
28	22	eldifad	$$⊢\phi \to Y\in V$$
29	6 18	lmodvacl	$$⊢\left(U\in \mathrm{LMod}\wedge w\in V\wedge Y\in V\right)\to w\underset{\dot{}}{+}Y\in V$$
30	26 27 28 29	syl3anc	$$⊢\phi \to w\underset{\dot{}}{+}Y\in V$$
31	3 5 14	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
32	17	eldifad	$$⊢\phi \to X\in V$$
33	6 9 31 27 32 28 25	lspindpi	$$⊢\phi \to \left(N\left(\left\{w\right\}\right)\ne N\left(\left\{X\right\}\right)\wedge N\left(\left\{w\right\}\right)\ne N\left(\left\{Y\right\}\right)\right)$$
34	33	simprd	$$⊢\phi \to N\left(\left\{w\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
35	6 18 8 9 26 27 28 34	lmodindp1	$$⊢\phi \to w\underset{\dot{}}{+}Y\ne \underset{\dot{}}{0}$$
36		eldifsn	$$⊢w\underset{\dot{}}{+}Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(w\underset{\dot{}}{+}Y\in V\wedge w\underset{\dot{}}{+}Y\ne \underset{\dot{}}{0}\right)$$
37	30 35 36	sylanbrc	$$⊢\phi \to w\underset{\dot{}}{+}Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
38	23	eldifad	$$⊢\phi \to Z\in V$$
39	6 9 31 32 28 38 20	lspindpi	$$⊢\phi \to \left(N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)\wedge N\left(\left\{X\right\}\right)\ne N\left(\left\{Z\right\}\right)\right)$$
40	39	simpld	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
41	6 18 8 9 31 17 22 23 24 21 40 25	mapdindp3	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{w\underset{\dot{}}{+}Y\right\}\right)$$
42	6 18 8 9 31 17 22 23 24 21 40 25	mapdindp4	$$⊢\phi \to \neg Z\in N\left(\left\{X,w\underset{\dot{}}{+}Y\right\}\right)$$
43	6 8 9 31 17 30 38 41 42	lspindp1	$$⊢\phi \to \left(N\left(\left\{Z\right\}\right)\ne N\left(\left\{w\underset{\dot{}}{+}Y\right\}\right)\wedge \neg X\in N\left(\left\{Z,w\underset{\dot{}}{+}Y\right\}\right)\right)$$
44	43	simprd	$$⊢\phi \to \neg X\in N\left(\left\{Z,w\underset{\dot{}}{+}Y\right\}\right)$$
45		prcom	$$⊢\left\{w\underset{\dot{}}{+}Y,Z\right\}=\left\{Z,w\underset{\dot{}}{+}Y\right\}$$
46	45	fveq2i	$$⊢N\left(\left\{w\underset{\dot{}}{+}Y,Z\right\}\right)=N\left(\left\{Z,w\underset{\dot{}}{+}Y\right\}\right)$$
47	46	eleq2i	$$⊢X\in N\left(\left\{w\underset{\dot{}}{+}Y,Z\right\}\right)\leftrightarrow X\in N\left(\left\{Z,w\underset{\dot{}}{+}Y\right\}\right)$$
48	44 47	sylnibr	$$⊢\phi \to \neg X\in N\left(\left\{w\underset{\dot{}}{+}Y,Z\right\}\right)$$
49	6 9 31 38 32 30 42	lspindpi	$$⊢\phi \to \left(N\left(\left\{Z\right\}\right)\ne N\left(\left\{X\right\}\right)\wedge N\left(\left\{Z\right\}\right)\ne N\left(\left\{w\underset{\dot{}}{+}Y\right\}\right)\right)$$
50	49	simprd	$$⊢\phi \to N\left(\left\{Z\right\}\right)\ne N\left(\left\{w\underset{\dot{}}{+}Y\right\}\right)$$
51	50	necomd	$$⊢\phi \to N\left(\left\{w\underset{\dot{}}{+}Y\right\}\right)\ne N\left(\left\{Z\right\}\right)$$
52		eqidd	$$⊢\phi \to I\left(⟨X,F,w\underset{\dot{}}{+}Y⟩\right)=I\left(⟨X,F,w\underset{\dot{}}{+}Y⟩\right)$$
53		eqidd	$$⊢\phi \to I\left(⟨X,F,Z⟩\right)=I\left(⟨X,F,Z⟩\right)$$
54	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 37 23 48 51 52 53	mapdh6aN	$$⊢\phi \to I\left(⟨X,F,\left(w\underset{\dot{}}{+}Y\right)\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,w\underset{\dot{}}{+}Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$