mapdh6iN

Description: Lemmma for mapdh6N . Eliminate auxiliary vector w . (Contributed by NM, 1-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mapdh.q	$⊢ Q = 0_{C}$
	mapdh.i	$⊢ I = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
	mapdh.h	$⊢ H = LHyp (K)$
	mapdh.m	$⊢ M = mapd (K) (W)$
	mapdh.u	$⊢ U = DVecH (K) (W)$
	mapdh.v	$⊢ V = {Base}_{U}$
	mapdh.s	$⊢ \underset{˙}{-} = -_{U}$
	mapdhc.o	$⊢ \underset{˙}{0} = 0_{U}$
	mapdh.n	$⊢ N = LSpan (U)$
	mapdh.c	$⊢ C = LCDual (K) (W)$
	mapdh.d	$⊢ D = {Base}_{C}$
	mapdh.r	$⊢ R = -_{C}$
	mapdh.j	$⊢ J = LSpan (C)$
	mapdh.k	$⊢ φ \to (K \in HL \land W \in H)$
	mapdhc.f	$⊢ φ \to F \in D$
	mapdh.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
	mapdhcl.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
	mapdh.p	$⊢ \underset{˙}{+} = +_{U}$
	mapdh.a	$⊢ \underset{˙}{✚} = +_{C}$
	mapdh6i.xn	$⊢ φ \to \neg X \in N (\{Y, Z\})$
	mapdh6i.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdh6i.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
	mapdh6i.yz	$⊢ φ \to N (\{Y\}) = N (\{Z\})$
Assertion	mapdh6iN	$⊢ φ \to I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)$

Proof

Step	Hyp	Ref	Expression
1		mapdh.q	$⊢ Q = 0_{C}$
2		mapdh.i	$⊢ I = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
3		mapdh.h	$⊢ H = LHyp (K)$
4		mapdh.m	$⊢ M = mapd (K) (W)$
5		mapdh.u	$⊢ U = DVecH (K) (W)$
6		mapdh.v	$⊢ V = {Base}_{U}$
7		mapdh.s	$⊢ \underset{˙}{-} = -_{U}$
8		mapdhc.o	$⊢ \underset{˙}{0} = 0_{U}$
9		mapdh.n	$⊢ N = LSpan (U)$
10		mapdh.c	$⊢ C = LCDual (K) (W)$
11		mapdh.d	$⊢ D = {Base}_{C}$
12		mapdh.r	$⊢ R = -_{C}$
13		mapdh.j	$⊢ J = LSpan (C)$
14		mapdh.k	$⊢ φ \to (K \in HL \land W \in H)$
15		mapdhc.f	$⊢ φ \to F \in D$
16		mapdh.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
17		mapdhcl.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
18		mapdh.p	$⊢ \underset{˙}{+} = +_{U}$
19		mapdh.a	$⊢ \underset{˙}{✚} = +_{C}$
20		mapdh6i.xn	$⊢ φ \to \neg X \in N (\{Y, Z\})$
21		mapdh6i.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
22		mapdh6i.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
23		mapdh6i.yz	$⊢ φ \to N (\{Y\}) = N (\{Z\})$
24	17	eldifad	$⊢ φ \to X \in V$
25	21	eldifad	$⊢ φ \to Y \in V$
26	3 5 6 9 14 24 25	dvh3dim	$⊢ φ \to \exists w \in V \neg w \in N (\{X, Y\})$
27	14	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to (K \in HL \land W \in H)$
28	15	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to F \in D$
29	16	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to M (N (\{X\})) = J (\{F\})$
30	17	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to X \in (V ∖ \{\underset{˙}{0}\})$
31	20	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to \neg X \in N (\{Y, Z\})$
32	23	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to N (\{Y\}) = N (\{Z\})$
33	21	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to Y \in (V ∖ \{\underset{˙}{0}\})$
34	22	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to Z \in (V ∖ \{\underset{˙}{0}\})$
35		eqid	$⊢ LSubSp (U) = LSubSp (U)$
36	3 5 14	dvhlmod	$⊢ φ \to U \in LMod$
37	36	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to U \in LMod$
38	6 35 9 36 24 25	lspprcl	$⊢ φ \to N (\{X, Y\}) \in LSubSp (U)$
39	38	3ad2ant1	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to N (\{X, Y\}) \in LSubSp (U)$
40		simp2	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to w \in V$
41		simp3	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to \neg w \in N (\{X, Y\})$
42	8 35 37 39 40 41	lssneln0	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to w \in (V ∖ \{\underset{˙}{0}\})$
43	1 2 3 4 5 6 7 8 9 10 11 12 13 27 28 29 30 18 19 31 32 33 34 42 41	mapdh6hN	$⊢ (φ \land w \in V \land \neg w \in N (\{X, Y\})) \to I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)$
44	43	rexlimdv3a	$⊢ φ \to (\exists w \in V \neg w \in N (\{X, Y\}) \to I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))$
45	26 44	mpd	$⊢ φ \to I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)$

Metamath Proof Explorer

Theorem mapdh6iN

Proof