mapdh6hN

Description: Lemmma for mapdh6N . Part (6) of Baer p. 48 line 2. (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	$⊢ \underset{˙}{✚} = +_{C}$
	mapdh6d.xn	$⊢ φ \to \neg X \in N (\{Y, Z\})$
	mapdh6d.yz	$⊢ φ \to N (\{Y\}) = N (\{Z\})$
	mapdh6d.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdh6d.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
	mapdh6d.w	$⊢ φ \to w \in (V ∖ \{\underset{˙}{0}\})$
	mapdh6d.wn	$⊢ φ \to \neg w \in N (\{X, Y\})$
Assertion	mapdh6hN	$⊢ φ \to I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)$

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		mapdh6d.xn	$⊢ φ \to \neg X \in N (\{Y, Z\})$
21		mapdh6d.yz	$⊢ φ \to N (\{Y\}) = N (\{Z\})$
22		mapdh6d.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
23		mapdh6d.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
24		mapdh6d.w	$⊢ φ \to w \in (V ∖ \{\underset{˙}{0}\})$
25		mapdh6d.wn	$⊢ φ \to \neg w \in N (\{X, Y\})$
26	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25	mapdh6gN	$⊢ φ \to I (⟨X, F, w⟩) \underset{˙}{✚} I (⟨X, F, Y \underset{˙}{+} Z⟩) = (I (⟨X, F, w⟩) \underset{˙}{✚} I (⟨X, F, Y⟩)) \underset{˙}{✚} I (⟨X, F, Z⟩)$
27	3 10 14	lcdlmod	$⊢ φ \to C \in LMod$
28	24	eldifad	$⊢ φ \to w \in V$
29	3 5 14	dvhlvec	$⊢ φ \to U \in LVec$
30	17	eldifad	$⊢ φ \to X \in V$
31	22	eldifad	$⊢ φ \to Y \in V$
32	6 9 29 28 30 31 25	lspindpi	$⊢ φ \to (N (\{w\}) \neq N (\{X\}) \land N (\{w\}) \neq N (\{Y\}))$
33	32	simpld	$⊢ φ \to N (\{w\}) \neq N (\{X\})$
34	33	necomd	$⊢ φ \to N (\{X\}) \neq N (\{w\})$
35	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 28 34	mapdhcl	$⊢ φ \to I (⟨X, F, w⟩) \in D$
36	23	eldifad	$⊢ φ \to Z \in V$
37	6 9 29 30 31 36 20	lspindpi	$⊢ φ \to (N (\{X\}) \neq N (\{Y\}) \land N (\{X\}) \neq N (\{Z\}))$
38	37	simpld	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
39	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 31 38	mapdhcl	$⊢ φ \to I (⟨X, F, Y⟩) \in D$
40	37	simprd	$⊢ φ \to N (\{X\}) \neq N (\{Z\})$
41	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 36 40	mapdhcl	$⊢ φ \to I (⟨X, F, Z⟩) \in D$
42	11 19	lmodass	$⊢ (C \in LMod \land (I (⟨X, F, w⟩) \in D \land I (⟨X, F, Y⟩) \in D \land I (⟨X, F, Z⟩) \in D)) \to (I (⟨X, F, w⟩) \underset{˙}{✚} I (⟨X, F, Y⟩)) \underset{˙}{✚} I (⟨X, F, Z⟩) = I (⟨X, F, w⟩) \underset{˙}{✚} (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))$
43	27 35 39 41 42	syl13anc	$⊢ φ \to (I (⟨X, F, w⟩) \underset{˙}{✚} I (⟨X, F, Y⟩)) \underset{˙}{✚} I (⟨X, F, Z⟩) = I (⟨X, F, w⟩) \underset{˙}{✚} (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))$
44	26 43	eqtrd	$⊢ φ \to I (⟨X, F, w⟩) \underset{˙}{✚} I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, w⟩) \underset{˙}{✚} (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))$
45	3 5 14	dvhlmod	$⊢ φ \to U \in LMod$
46	6 18	lmodvacl	$⊢ (U \in LMod \land Y \in V \land Z \in V) \to Y \underset{˙}{+} Z \in V$
47	45 31 36 46	syl3anc	$⊢ φ \to Y \underset{˙}{+} Z \in V$
48	6 18 8 9 29 17 22 23 24 21 38 25	mapdindp1	$⊢ φ \to N (\{X\}) \neq N (\{Y \underset{˙}{+} Z\})$
49	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 47 48	mapdhcl	$⊢ φ \to I (⟨X, F, Y \underset{˙}{+} Z⟩) \in D$
50	11 19	lmodvacl	$⊢ (C \in LMod \land I (⟨X, F, Y⟩) \in D \land I (⟨X, F, Z⟩) \in D) \to I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩) \in D$
51	27 39 41 50	syl3anc	$⊢ φ \to I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩) \in D$
52	11 19	lmodlcan	$⊢ (C \in LMod \land (I (⟨X, F, Y \underset{˙}{+} Z⟩) \in D \land I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩) \in D \land I (⟨X, F, w⟩) \in D)) \to (I (⟨X, F, w⟩) \underset{˙}{✚} I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, w⟩) \underset{˙}{✚} (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)) \leftrightarrow I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))$
53	27 49 51 35 52	syl13anc	$⊢ φ \to (I (⟨X, F, w⟩) \underset{˙}{✚} I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, w⟩) \underset{˙}{✚} (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)) \leftrightarrow I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))$
54	44 53	mpbid	$⊢ φ \to I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)$

Metamath Proof Explorer

Theorem mapdh6hN

Proof