mapdh8ac

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 13-May-2015)

	Ref	Expression
Hypotheses	mapdh8a.h	$⊢ H = LHyp (K)$
	mapdh8a.u	$⊢ U = DVecH (K) (W)$
	mapdh8a.v	$⊢ V = {Base}_{U}$
	mapdh8a.s	$⊢ \underset{˙}{-} = -_{U}$
	mapdh8a.o	$⊢ \underset{˙}{0} = 0_{U}$
	mapdh8a.n	$⊢ N = LSpan (U)$
	mapdh8a.c	$⊢ C = LCDual (K) (W)$
	mapdh8a.d	$⊢ D = {Base}_{C}$
	mapdh8a.r	$⊢ R = -_{C}$
	mapdh8a.q	$⊢ Q = 0_{C}$
	mapdh8a.j	$⊢ J = LSpan (C)$
	mapdh8a.m	$⊢ M = mapd (K) (W)$
	mapdh8a.i	$⊢ I = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
	mapdh8a.k	$⊢ φ \to (K \in HL \land W \in H)$
	mapdh8ac.f	$⊢ φ \to F \in D$
	mapdh8ac.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
	mapdh8ac.eg	$⊢ φ \to I (⟨X, F, Y⟩) = G$
	mapdh8ac.ee	$⊢ φ \to I (⟨X, F, Z⟩) = E$
	mapdh8ac.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8ac.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8ac.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8ac.t	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8ac.yn	$⊢ φ \to N (\{X\}) = N (\{T\})$
	mapdh8ac.ew	$⊢ φ \to I (⟨X, F, w⟩) = B$
	mapdh8ac.w	$⊢ φ \to w \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8ac.yw	$⊢ φ \to N (\{Y\}) \neq N (\{w\})$
	mapdh8ac.xy	$⊢ φ \to \neg X \in N (\{Y, w\})$
	mapdh8ac.wz	$⊢ φ \to N (\{w\}) \neq N (\{Z\})$
	mapdh8ac.xz	$⊢ φ \to \neg X \in N (\{w, Z\})$
Assertion	mapdh8ac	$⊢ φ \to I (⟨Y, G, T⟩) = I (⟨Z, E, T⟩)$

Proof

Step	Hyp	Ref	Expression
1		mapdh8a.h	$⊢ H = LHyp (K)$
2		mapdh8a.u	$⊢ U = DVecH (K) (W)$
3		mapdh8a.v	$⊢ V = {Base}_{U}$
4		mapdh8a.s	$⊢ \underset{˙}{-} = -_{U}$
5		mapdh8a.o	$⊢ \underset{˙}{0} = 0_{U}$
6		mapdh8a.n	$⊢ N = LSpan (U)$
7		mapdh8a.c	$⊢ C = LCDual (K) (W)$
8		mapdh8a.d	$⊢ D = {Base}_{C}$
9		mapdh8a.r	$⊢ R = -_{C}$
10		mapdh8a.q	$⊢ Q = 0_{C}$
11		mapdh8a.j	$⊢ J = LSpan (C)$
12		mapdh8a.m	$⊢ M = mapd (K) (W)$
13		mapdh8a.i	$⊢ I = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
14		mapdh8a.k	$⊢ φ \to (K \in HL \land W \in H)$
15		mapdh8ac.f	$⊢ φ \to F \in D$
16		mapdh8ac.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
17		mapdh8ac.eg	$⊢ φ \to I (⟨X, F, Y⟩) = G$
18		mapdh8ac.ee	$⊢ φ \to I (⟨X, F, Z⟩) = E$
19		mapdh8ac.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
20		mapdh8ac.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
21		mapdh8ac.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
22		mapdh8ac.t	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
23		mapdh8ac.yn	$⊢ φ \to N (\{X\}) = N (\{T\})$
24		mapdh8ac.ew	$⊢ φ \to I (⟨X, F, w⟩) = B$
25		mapdh8ac.w	$⊢ φ \to w \in (V ∖ \{\underset{˙}{0}\})$
26		mapdh8ac.yw	$⊢ φ \to N (\{Y\}) \neq N (\{w\})$
27		mapdh8ac.xy	$⊢ φ \to \neg X \in N (\{Y, w\})$
28		mapdh8ac.wz	$⊢ φ \to N (\{w\}) \neq N (\{Z\})$
29		mapdh8ac.xz	$⊢ φ \to \neg X \in N (\{w, Z\})$
30	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 24 19 20 25 22 26 27 23	mapdh8ab	$⊢ φ \to I (⟨Y, G, T⟩) = I (⟨w, B, T⟩)$
31	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 24 18 19 25 21 22 28 29 23	mapdh8ab	$⊢ φ \to I (⟨w, B, T⟩) = I (⟨Z, E, T⟩)$
32	30 31	eqtrd	$⊢ φ \to I (⟨Y, G, T⟩) = I (⟨Z, E, T⟩)$

Metamath Proof Explorer

Theorem mapdh8ac

Proof