Theorem mapdh6bN

Description: Lemmma for mapdh6N . (Contributed by NM, 24-Apr-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	$$⊢\underset{\dot{}}{✚}={+}_C$$
	mapdh6b.y	$$⊢\phi \to Y=\underset{\dot{}}{0}$$
	mapdh6b.z	$$⊢\phi \to Z\in V$$
	mapdh6b.ne	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$
Assertion	mapdh6bN	$$⊢\phi \to I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$

Proof

Step	Hyp	Ref	Expression
1		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		mapdh6b.y	$$⊢\phi \to Y=\underset{\dot{}}{0}$$
21		mapdh6b.z	$$⊢\phi \to Z\in V$$
22		mapdh6b.ne	$$⊢\phi \to \neg X\in N\left(\left\{Y,Z\right\}\right)$$
23	3 10 14	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	3 5 14	dvhlvec	$$⊢\phi \to U\in \mathrm{LVec}$$
27	17	eldifad	$$⊢\phi \to X\in V$$
28	3 5 14	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
29	6 8	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	6 9 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 4 5 6 7 8 9 10 11 12 13 14 15 16 17 21 33	mapdhcl	$$⊢\phi \to I\left(⟨X,F,Z⟩\right)\in D$$
35	11 19 1	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 8 17 15	mapdhval0	$$⊢\phi \to I\left(⟨X,F,\underset{\dot{}}{0}⟩\right)=Q$$
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	6 18 8	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	$$⊢\phi \to I\left(⟨X,F,Y\underset{\dot{}}{+}Z⟩\right)=I\left(⟨X,F,Y⟩\right)\underset{\dot{}}{✚}I\left(⟨X,F,Z⟩\right)$$