Theorem mapdheq2biN

Description: Lemmma for ~? mapdh . Part (2) in Baer p. 45. The bidirectional version of mapdheq2 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-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	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdhe2.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	mapdhe2.g	$$⊢\phi \to G\in D$$
	mapdh.ne3	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
	mapdh.my	$$⊢\phi \to M\left(N\left(\left\{Y\right\}\right)\right)=J\left(\left\{G\right\}\right)$$
Assertion	mapdheq2biN	$$⊢\phi \to \left(I\left(⟨X,F,Y⟩\right)=G\leftrightarrow I\left(⟨Y,G,X⟩\right)=F\right)$$

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		mapdhe2.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
19		mapdhe2.g	$$⊢\phi \to G\in D$$
20		mapdh.ne3	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
21		mapdh.my	$$⊢\phi \to M\left(N\left(\left\{Y\right\}\right)\right)=J\left(\left\{G\right\}\right)$$
22	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	mapdheq2	$$⊢\phi \to \left(I\left(⟨X,F,Y⟩\right)=G\to I\left(⟨Y,G,X⟩\right)=F\right)$$
23	20	necomd	$$⊢\phi \to N\left(\left\{Y\right\}\right)\ne N\left(\left\{X\right\}\right)$$
24	1 2 3 4 5 6 7 8 9 10 11 12 13 14 19 21 18 17 15 23	mapdheq2	$$⊢\phi \to \left(I\left(⟨Y,G,X⟩\right)=F\to I\left(⟨X,F,Y⟩\right)=G\right)$$
25	22 24	impbid	$$⊢\phi \to \left(I\left(⟨X,F,Y⟩\right)=G\leftrightarrow I\left(⟨Y,G,X⟩\right)=F\right)$$