mapdh8j

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	$⊢ φ \to (K \in HL \land W \in H)$
	mapdh8h.f	$⊢ φ \to F \in D$
	mapdh8h.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
	mapdh8i.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8i.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8i.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8i.xy	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
	mapdh8i.xz	$⊢ φ \to N (\{X\}) \neq N (\{Z\})$
	mapdh8i.yt	$⊢ φ \to N (\{Y\}) \neq N (\{T\})$
	mapdh8i.zt	$⊢ φ \to N (\{Z\}) \neq N (\{T\})$
	mapdh8j.t	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
Assertion	mapdh8j	$⊢ φ \to I (⟨Y, I (⟨X, F, Y⟩), T⟩) = I (⟨Z, I (⟨X, F, Z⟩), T⟩)$

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		mapdh8h.f	$⊢ φ \to F \in D$
16		mapdh8h.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
17		mapdh8i.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
18		mapdh8i.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
19		mapdh8i.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
20		mapdh8i.xy	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
21		mapdh8i.xz	$⊢ φ \to N (\{X\}) \neq N (\{Z\})$
22		mapdh8i.yt	$⊢ φ \to N (\{Y\}) \neq N (\{T\})$
23		mapdh8i.zt	$⊢ φ \to N (\{Z\}) \neq N (\{T\})$
24		mapdh8j.t	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
25	14	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to (K \in HL \land W \in H)$
26	15	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to F \in D$
27	16	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to M (N (\{X\})) = J (\{F\})$
28		eqidd	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to I (⟨X, F, Y⟩) = I (⟨X, F, Y⟩)$
29		eqidd	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to I (⟨X, F, Z⟩) = I (⟨X, F, Z⟩)$
30	17	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to X \in (V ∖ \{\underset{˙}{0}\})$
31	18	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to Y \in (V ∖ \{\underset{˙}{0}\})$
32	19	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to Z \in (V ∖ \{\underset{˙}{0}\})$
33	24	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to T \in (V ∖ \{\underset{˙}{0}\})$
34		simpr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to N (\{X\}) = N (\{T\})$
35	20	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to N (\{X\}) \neq N (\{Y\})$
36	21	adantr	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to N (\{X\}) \neq N (\{Z\})$
37	1 2 3 4 5 6 7 8 9 10 11 12 13 25 26 27 28 29 30 31 32 33 34 35 36	mapdh8ad	$⊢ (φ \land N (\{X\}) = N (\{T\})) \to I (⟨Y, I (⟨X, F, Y⟩), T⟩) = I (⟨Z, I (⟨X, F, Z⟩), T⟩)$
38	14	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to (K \in HL \land W \in H)$
39	15	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to F \in D$
40	16	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to M (N (\{X\})) = J (\{F\})$
41	17	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to X \in (V ∖ \{\underset{˙}{0}\})$
42	18	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to Y \in (V ∖ \{\underset{˙}{0}\})$
43	19	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to Z \in (V ∖ \{\underset{˙}{0}\})$
44	20	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to N (\{X\}) \neq N (\{Y\})$
45	21	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to N (\{X\}) \neq N (\{Z\})$
46	22	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to N (\{Y\}) \neq N (\{T\})$
47	23	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to N (\{Z\}) \neq N (\{T\})$
48	24	adantr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to T \in (V ∖ \{\underset{˙}{0}\})$
49		simpr	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to N (\{X\}) \neq N (\{T\})$
50	1 2 3 4 5 6 7 8 9 10 11 12 13 38 39 40 41 42 43 44 45 46 47 48 49	mapdh8i	$⊢ (φ \land N (\{X\}) \neq N (\{T\})) \to I (⟨Y, I (⟨X, F, Y⟩), T⟩) = I (⟨Z, I (⟨X, F, Z⟩), T⟩)$
51	37 50	pm2.61dane	$⊢ φ \to I (⟨Y, I (⟨X, F, Y⟩), T⟩) = I (⟨Z, I (⟨X, F, Z⟩), T⟩)$

Metamath Proof Explorer

Theorem mapdh8j

Proof