mapdh8c

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 6-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	$⊢ φ \to (K \in HL \land W \in H)$
	mapdh8c.f	$⊢ φ \to F \in D$
	mapdh8c.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
	mapdh8c.a	$⊢ φ \to I (⟨X, F, w⟩) = E$
	mapdh8c.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8c.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8c.xt	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8c.yz	$⊢ φ \to N (\{Y\}) \neq N (\{T\})$
	mapdh8c.w	$⊢ φ \to w \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8c.wt	$⊢ φ \to N (\{w\}) \neq N (\{T\})$
	mapdh8c.ut	$⊢ φ \to N (\{X\}) \neq N (\{T\})$
	mapdh8c.vw	$⊢ φ \to N (\{Y\}) \neq N (\{w\})$
	mapdh8c.e	$⊢ φ \to X \in N (\{Y, T\})$
	mapdh8c.xn	$⊢ φ \to \neg X \in N (\{Y, w\})$
Assertion	mapdh8c	$⊢ φ \to I (⟨w, E, T⟩) = I (⟨X, F, T⟩)$

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		mapdh8c.f	$⊢ φ \to F \in D$
16		mapdh8c.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
17		mapdh8c.a	$⊢ φ \to I (⟨X, F, w⟩) = E$
18		mapdh8c.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
19		mapdh8c.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
20		mapdh8c.xt	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
21		mapdh8c.yz	$⊢ φ \to N (\{Y\}) \neq N (\{T\})$
22		mapdh8c.w	$⊢ φ \to w \in (V ∖ \{\underset{˙}{0}\})$
23		mapdh8c.wt	$⊢ φ \to N (\{w\}) \neq N (\{T\})$
24		mapdh8c.ut	$⊢ φ \to N (\{X\}) \neq N (\{T\})$
25		mapdh8c.vw	$⊢ φ \to N (\{Y\}) \neq N (\{w\})$
26		mapdh8c.e	$⊢ φ \to X \in N (\{Y, T\})$
27		mapdh8c.xn	$⊢ φ \to \neg X \in N (\{Y, w\})$
28	1 2 14	dvhlvec	$⊢ φ \to U \in LVec$
29	18	eldifad	$⊢ φ \to X \in V$
30	19	eldifad	$⊢ φ \to Y \in V$
31	22	eldifad	$⊢ φ \to w \in V$
32	3 6 28 29 30 31 27	lspindpi	$⊢ φ \to (N (\{X\}) \neq N (\{Y\}) \land N (\{X\}) \neq N (\{w\}))$
33	32	simprd	$⊢ φ \to N (\{X\}) \neq N (\{w\})$
34	20	eldifad	$⊢ φ \to T \in V$
35	3 5 6 28 18 30 34 24 26	lspexch	$⊢ φ \to Y \in N (\{X, T\})$
36	28	adantr	$⊢ (φ \land Y \in N (\{X, w\})) \to U \in LVec$
37	19	adantr	$⊢ (φ \land Y \in N (\{X, w\})) \to Y \in (V ∖ \{\underset{˙}{0}\})$
38	29	adantr	$⊢ (φ \land Y \in N (\{X, w\})) \to X \in V$
39	31	adantr	$⊢ (φ \land Y \in N (\{X, w\})) \to w \in V$
40	25	adantr	$⊢ (φ \land Y \in N (\{X, w\})) \to N (\{Y\}) \neq N (\{w\})$
41		simpr	$⊢ (φ \land Y \in N (\{X, w\})) \to Y \in N (\{X, w\})$
42	3 5 6 36 37 38 39 40 41	lspexch	$⊢ (φ \land Y \in N (\{X, w\})) \to X \in N (\{Y, w\})$
43	27 42	mtand	$⊢ φ \to \neg Y \in N (\{X, w\})$
44	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 22 23 20 33 35 43	mapdh8b	$⊢ φ \to I (⟨w, E, T⟩) = I (⟨X, F, T⟩)$

Metamath Proof Explorer

Theorem mapdh8c

Proof