mapdh8b

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	$⊢ φ \to (K \in HL \land W \in H)$
	mapdh8b.f	$⊢ φ \to G \in D$
	mapdh8b.mn	$⊢ φ \to M (N (\{Y\})) = J (\{G\})$
	mapdh8b.a	$⊢ φ \to I (⟨Y, G, w⟩) = E$
	mapdh8b.x	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8b.y	$⊢ φ \to w \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8b.yz	$⊢ φ \to N (\{w\}) \neq N (\{T\})$
	mapdh8b.xt	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8b.vw	$⊢ φ \to N (\{Y\}) \neq N (\{w\})$
	mapdh8b.e	$⊢ φ \to X \in N (\{Y, T\})$
	mapdh8b.xn	$⊢ φ \to \neg X \in N (\{Y, w\})$
Assertion	mapdh8b	$⊢ φ \to I (⟨w, E, T⟩) = I (⟨Y, G, T⟩)$

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		mapdh8b.f	$⊢ φ \to G \in D$
16		mapdh8b.mn	$⊢ φ \to M (N (\{Y\})) = J (\{G\})$
17		mapdh8b.a	$⊢ φ \to I (⟨Y, G, w⟩) = E$
18		mapdh8b.x	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
19		mapdh8b.y	$⊢ φ \to w \in (V ∖ \{\underset{˙}{0}\})$
20		mapdh8b.yz	$⊢ φ \to N (\{w\}) \neq N (\{T\})$
21		mapdh8b.xt	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
22		mapdh8b.vw	$⊢ φ \to N (\{Y\}) \neq N (\{w\})$
23		mapdh8b.e	$⊢ φ \to X \in N (\{Y, T\})$
24		mapdh8b.xn	$⊢ φ \to \neg X \in N (\{Y, w\})$
25	1 2 14	dvhlvec	$⊢ φ \to U \in LVec$
26	18	eldifad	$⊢ φ \to Y \in V$
27	19	eldifad	$⊢ φ \to w \in V$
28	21	eldifad	$⊢ φ \to T \in V$
29	3 6 25 26 27 28 23 24	lspindp5	$⊢ φ \to \neg T \in N (\{Y, w\})$
30		prcom	$⊢ \{w, T\} = \{T, w\}$
31	30	fveq2i	$⊢ N (\{w, T\}) = N (\{T, w\})$
32	31	eleq2i	$⊢ Y \in N (\{w, T\}) \leftrightarrow Y \in N (\{T, w\})$
33	25	adantr	$⊢ (φ \land Y \in N (\{T, w\})) \to U \in LVec$
34	18	adantr	$⊢ (φ \land Y \in N (\{T, w\})) \to Y \in (V ∖ \{\underset{˙}{0}\})$
35	28	adantr	$⊢ (φ \land Y \in N (\{T, w\})) \to T \in V$
36	27	adantr	$⊢ (φ \land Y \in N (\{T, w\})) \to w \in V$
37	22	adantr	$⊢ (φ \land Y \in N (\{T, w\})) \to N (\{Y\}) \neq N (\{w\})$
38		simpr	$⊢ (φ \land Y \in N (\{T, w\})) \to Y \in N (\{T, w\})$
39	3 5 6 33 34 35 36 37 38	lspexch	$⊢ (φ \land Y \in N (\{T, w\})) \to T \in N (\{Y, w\})$
40	39	ex	$⊢ φ \to (Y \in N (\{T, w\}) \to T \in N (\{Y, w\}))$
41	32 40	syl5bi	$⊢ φ \to (Y \in N (\{w, T\}) \to T \in N (\{Y, w\}))$
42	29 41	mtod	$⊢ φ \to \neg Y \in N (\{w, T\})$
43	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 42	mapdh8a	$⊢ φ \to I (⟨w, E, T⟩) = I (⟨Y, G, T⟩)$

Metamath Proof Explorer

Theorem mapdh8b

Proof