mapdh6aN

Description: Lemma for mapdh6N . Part (6) in Baer p. 47, case 1. (Contributed by NM, 23-Apr-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	$⊢ \underset{˙}{✚} = +_{C}$
	mapdhe6.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdhe6.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
	mapdhe6.xn	$⊢ φ \to \neg X \in N (\{Y, Z\})$
	mapdh6.yz	$⊢ φ \to N (\{Y\}) \neq N (\{Z\})$
	mapdh6.fg	$⊢ φ \to I (⟨X, F, Y⟩) = G$
	mapdh6.fe	$⊢ φ \to I (⟨X, F, Z⟩) = E$
Assertion	mapdh6aN	$⊢ φ \to I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)$

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		mapdhe6.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
21		mapdhe6.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
22		mapdhe6.xn	$⊢ φ \to \neg X \in N (\{Y, Z\})$
23		mapdh6.yz	$⊢ φ \to N (\{Y\}) \neq N (\{Z\})$
24		mapdh6.fg	$⊢ φ \to I (⟨X, F, Y⟩) = G$
25		mapdh6.fe	$⊢ φ \to I (⟨X, F, Z⟩) = E$
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	mapdh6lem2N	$⊢ φ \to M (N (\{Y \underset{˙}{+} Z\})) = J (\{G \underset{˙}{✚} E\})$
27	24 25	oveq12d	$⊢ φ \to I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩) = G \underset{˙}{✚} E$
28	27	sneqd	$⊢ φ \to \{I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)\} = \{G \underset{˙}{✚} E\}$
29	28	fveq2d	$⊢ φ \to J (\{I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)\}) = J (\{G \underset{˙}{✚} E\})$
30	26 29	eqtr4d	$⊢ φ \to M (N (\{Y \underset{˙}{+} Z\})) = J (\{I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)\})$
31	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	mapdh6lem1N	$⊢ φ \to M (N (\{X \underset{˙}{-} (Y \underset{˙}{+} Z)\})) = J (\{F R (G \underset{˙}{✚} E)\})$
32	27	oveq2d	$⊢ φ \to F R (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)) = F R (G \underset{˙}{✚} E)$
33	32	sneqd	$⊢ φ \to \{F R (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))\} = \{F R (G \underset{˙}{✚} E)\}$
34	33	fveq2d	$⊢ φ \to J (\{F R (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))\}) = J (\{F R (G \underset{˙}{✚} E)\})$
35	31 34	eqtr4d	$⊢ φ \to M (N (\{X \underset{˙}{-} (Y \underset{˙}{+} Z)\})) = J (\{F R (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))\})$
36	3 5 14	dvhlmod	$⊢ φ \to U \in LMod$
37	20	eldifad	$⊢ φ \to Y \in V$
38	21	eldifad	$⊢ φ \to Z \in V$
39	6 18	lmodvacl	$⊢ (U \in LMod \land Y \in V \land Z \in V) \to Y \underset{˙}{+} Z \in V$
40	36 37 38 39	syl3anc	$⊢ φ \to Y \underset{˙}{+} Z \in V$
41	6 18 8 9 36 37 38 23	lmodindp1	$⊢ φ \to Y \underset{˙}{+} Z \neq \underset{˙}{0}$
42		eldifsn	$⊢ Y \underset{˙}{+} Z \in (V ∖ \{\underset{˙}{0}\}) \leftrightarrow (Y \underset{˙}{+} Z \in V \land Y \underset{˙}{+} Z \neq \underset{˙}{0})$
43	40 41 42	sylanbrc	$⊢ φ \to Y \underset{˙}{+} Z \in (V ∖ \{\underset{˙}{0}\})$
44	3 10 14	lcdlmod	$⊢ φ \to C \in LMod$
45	3 5 14	dvhlvec	$⊢ φ \to U \in LVec$
46	17	eldifad	$⊢ φ \to X \in V$
47	6 8 9 45 37 21 46 23 22	lspindp2	$⊢ φ \to (N (\{X\}) \neq N (\{Y\}) \land \neg Z \in N (\{X, Y\}))$
48	47	simpld	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
49	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 37 48	mapdhcl	$⊢ φ \to I (⟨X, F, Y⟩) \in D$
50	6 8 9 45 20 38 46 23 22	lspindp1	$⊢ φ \to (N (\{X\}) \neq N (\{Z\}) \land \neg Y \in N (\{X, Z\}))$
51	50	simpld	$⊢ φ \to N (\{X\}) \neq N (\{Z\})$
52	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 38 51	mapdhcl	$⊢ φ \to I (⟨X, F, Z⟩) \in D$
53	11 19	lmodvacl	$⊢ (C \in LMod \land I (⟨X, F, Y⟩) \in D \land I (⟨X, F, Z⟩) \in D) \to I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩) \in D$
54	44 49 52 53	syl3anc	$⊢ φ \to I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩) \in D$
55		eqid	$⊢ LSubSp (U) = LSubSp (U)$
56	6 55 9 36 37 38	lspprcl	$⊢ φ \to N (\{Y, Z\}) \in LSubSp (U)$
57	6 18 9 36 37 38	lspprvacl	$⊢ φ \to Y \underset{˙}{+} Z \in N (\{Y, Z\})$
58	55 9 36 56 57	ellspsn5	$⊢ φ \to N (\{Y \underset{˙}{+} Z\}) \subseteq N (\{Y, Z\})$
59	6 55 9 36 56 46	ellspsn5b	$⊢ φ \to (X \in N (\{Y, Z\}) \leftrightarrow N (\{X\}) \subseteq N (\{Y, Z\}))$
60	22 59	mtbid	$⊢ φ \to \neg N (\{X\}) \subseteq N (\{Y, Z\})$
61		nssne2	$⊢ (N (\{Y \underset{˙}{+} Z\}) \subseteq N (\{Y, Z\}) \land \neg N (\{X\}) \subseteq N (\{Y, Z\})) \to N (\{Y \underset{˙}{+} Z\}) \neq N (\{X\})$
62	58 60 61	syl2anc	$⊢ φ \to N (\{Y \underset{˙}{+} Z\}) \neq N (\{X\})$
63	62	necomd	$⊢ φ \to N (\{X\}) \neq N (\{Y \underset{˙}{+} Z\})$
64	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 43 54 63	mapdheq	$⊢ φ \to (I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩) \leftrightarrow (M (N (\{Y \underset{˙}{+} Z\})) = J (\{I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)\}) \land M (N (\{X \underset{˙}{-} (Y \underset{˙}{+} Z)\})) = J (\{F R (I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩))\})))$
65	30 35 64	mpbir2and	$⊢ φ \to I (⟨X, F, Y \underset{˙}{+} Z⟩) = I (⟨X, F, Y⟩) \underset{˙}{✚} I (⟨X, F, Z⟩)$

Metamath Proof Explorer

Theorem mapdh6aN

Proof