Theorem mapdh75e

Description: Part (7) of Baer p. 48 line 10 (5 of 6 cases). X , Y , Z are Baer's u, v, w. (Note: Cases 1 of 6 and 2 of 6 are hypotheses mapdh75b here and mapdh75a in mapdh75cN .) (Contributed by NM, 2-May-2015)

	Ref	Expression
Hypotheses	mapdh75.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	mapdh75.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	mapdh75.v	$$⊢V={\mathrm{Base}}_U$$
	mapdh75.s	$$⊢\underset{\dot{}}{-}={-}_U$$
	mapdh75.o	$$⊢\underset{\dot{}}{0}=0_U$$
	mapdh75.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	mapdh75.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	mapdh75.d	$$⊢D={\mathrm{Base}}_C$$
	mapdh75.r	$$⊢R={-}_C$$
	mapdh75.q	$$⊢Q=0_C$$
	mapdh75.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
	mapdh75.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	mapdh75.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
	mapdh75.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	mapdh75.f	$$⊢\phi \to F\in D$$
	mapdh75.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{F\right\}\right)$$
	mapdh75b	$$⊢\phi \to I\left(⟨X,F,Z⟩\right)=E$$
	mapdh75e.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Z\right\}\right)$$
	mapdh75e.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdh75e.z	$$⊢\phi \to Z\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
Assertion	mapdh75e	$$⊢\phi \to I\left(⟨Z,E,X⟩\right)=F$$

Proof

Step	Hyp	Ref	Expression
1		mapdh75.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		mapdh75.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		mapdh75.v	$$⊢V={\mathrm{Base}}_U$$
4		mapdh75.s	$$⊢\underset{\dot{}}{-}={-}_U$$
5		mapdh75.o	$$⊢\underset{\dot{}}{0}=0_U$$
6		mapdh75.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
7		mapdh75.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
8		mapdh75.d	$$⊢D={\mathrm{Base}}_C$$
9		mapdh75.r	$$⊢R={-}_C$$
10		mapdh75.q	$$⊢Q=0_C$$
11		mapdh75.j	$$⊢J=\mathrm{LSpan}\left(C\right)$$
12		mapdh75.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
13		mapdh75.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		mapdh75.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
15		mapdh75.f	$$⊢\phi \to F\in D$$
16		mapdh75.mn	$$⊢\phi \to M\left(N\left(\left\{X\right\}\right)\right)=J\left(\left\{F\right\}\right)$$
17		mapdh75b	$$⊢\phi \to I\left(⟨X,F,Z⟩\right)=E$$
18		mapdh75e.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Z\right\}\right)$$
19		mapdh75e.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
20		mapdh75e.z	$$⊢\phi \to Z\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
21	20	eldifad	$$⊢\phi \to Z\in V$$
22	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 21 18	mapdhcl	$$⊢\phi \to I\left(⟨X,F,Z⟩\right)\in D$$
23	17 22	eqeltrrd	$$⊢\phi \to E\in D$$
24	10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 20 23 18	mapdheq2	$$⊢\phi \to \left(I\left(⟨X,F,Z⟩\right)=E\to I\left(⟨Z,E,X⟩\right)=F\right)$$
25	17 24	mpd	$$⊢\phi \to I\left(⟨Z,E,X⟩\right)=F$$