Theorem mapdh8a

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 5-May-2015)

	Ref	Expression
Hypotheses	mapdh8a.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	mapdh8a.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	mapdh8a.v	$$⊢V={\mathrm{Base}}_U$$
	mapdh8a.s	$$⊢\underset{\dot{}}{-}={-}_U$$
	mapdh8a.o	$$⊢\underset{\dot{}}{0}=0_U$$
	mapdh8a.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	mapdh8a.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	mapdh8a.d	$$⊢D={\mathrm{Base}}_C$$
	mapdh8a.r	$$⊢R={-}_C$$
	mapdh8a.q	$$⊢Q=0_C$$
	mapdh8a.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
	mapdh8a.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	mapdh8a.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
	mapdh8a.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	mapdh8a.f	$$⊢\phi \to F\in D$$
	mapdh8a.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{F\right\}\right)$$
	mapdh8a.a	$$⊢\phi \to I\left(⟨X,F,Y⟩\right)=G$$
	mapdh8a.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh8a.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh8a.yz	$$⊢\phi \to N\left(\left\{Y\right\}\right)\ne N\left(\left\{T\right\}\right)$$
	mapdh8a.xt	$$⊢\phi \to T\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh8a.xn	$$⊢\phi \to \neg X\in N\left(\left\{Y,T\right\}\right)$$
Assertion	mapdh8a	$$⊢\phi \to I\left(⟨Y,G,T⟩\right)=I\left(⟨X,F,T⟩\right)$$

Proof

Step	Hyp	Ref	Expression
1		mapdh8a.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		mapdh8a.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		mapdh8a.v	$$⊢V={\mathrm{Base}}_U$$
4		mapdh8a.s	$$⊢\underset{\dot{}}{-}={-}_U$$
5		mapdh8a.o	$$⊢\underset{\dot{}}{0}=0_U$$
6		mapdh8a.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
7		mapdh8a.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
8		mapdh8a.d	$$⊢D={\mathrm{Base}}_C$$
9		mapdh8a.r	$$⊢R={-}_C$$
10		mapdh8a.q	$$⊢Q=0_C$$
11		mapdh8a.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
12		mapdh8a.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
13		mapdh8a.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
14		mapdh8a.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
15		mapdh8a.f	$$⊢\phi \to F\in D$$
16		mapdh8a.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{F\right\}\right)$$
17		mapdh8a.a	$$⊢\phi \to I\left(⟨X,F,Y⟩\right)=G$$
18		mapdh8a.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
19		mapdh8a.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
20		mapdh8a.yz	$$⊢\phi \to N\left(\left\{Y\right\}\right)\ne N\left(\left\{T\right\}\right)$$
21		mapdh8a.xt	$$⊢\phi \to T\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
22		mapdh8a.xn	$$⊢\phi \to \neg X\in N\left(\left\{Y,T\right\}\right)$$
23		eqidd	$$⊢\phi \to I\left(⟨X,F,T⟩\right)=I\left(⟨X,F,T⟩\right)$$
24	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 18 19 21 22 20 17 23	mapdheq4	$$⊢\phi \to I\left(⟨Y,G,T⟩\right)=I\left(⟨X,F,T⟩\right)$$