hdmap1l6lem1

Description: Lemma for hdmap1l6 . Part (6) in Baer p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015)

	Ref	Expression
Hypotheses	hdmap1l6.h	$⊢ H = LHyp (K)$
	hdmap1l6.u	$⊢ U = DVecH (K) (W)$
	hdmap1l6.v	$⊢ V = {Base}_{U}$
	hdmap1l6.p	$⊢ \underset{˙}{+} = +_{U}$
	hdmap1l6.s	$⊢ \underset{˙}{-} = -_{U}$
	hdmap1l6c.o	$⊢ \underset{˙}{0} = 0_{U}$
	hdmap1l6.n	$⊢ N = LSpan (U)$
	hdmap1l6.c	$⊢ C = LCDual (K) (W)$
	hdmap1l6.d	$⊢ D = {Base}_{C}$
	hdmap1l6.a	$⊢ \underset{˙}{✚} = +_{C}$
	hdmap1l6.r	$⊢ R = -_{C}$
	hdmap1l6.q	$⊢ Q = 0_{C}$
	hdmap1l6.l	$⊢ L = LSpan (C)$
	hdmap1l6.m	$⊢ M = mapd (K) (W)$
	hdmap1l6.i	$⊢ I = HDMap1 (K) (W)$
	hdmap1l6.k	$⊢ φ \to (K \in HL \land W \in H)$
	hdmap1l6.f	$⊢ φ \to F \in D$
	hdmap1l6cl.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
	hdmap1l6.mn	$⊢ φ \to M (N (\{X\})) = L (\{F\})$
	hdmap1l6e.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	hdmap1l6e.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
	hdmap1l6e.xn	$⊢ φ \to \neg X \in N (\{Y, Z\})$
	hdmap1l6.yz	$⊢ φ \to N (\{Y\}) \neq N (\{Z\})$
	hdmap1l6.fg	$⊢ φ \to I (⟨X, F, Y⟩) = G$
	hdmap1l6.fe	$⊢ φ \to I (⟨X, F, Z⟩) = E$
Assertion	hdmap1l6lem1	$⊢ φ \to M (N (\{X \underset{˙}{-} (Y \underset{˙}{+} Z)\})) = L (\{F R (G \underset{˙}{✚} E)\})$

Proof

Step	Hyp	Ref	Expression
1		hdmap1l6.h	$⊢ H = LHyp (K)$
2		hdmap1l6.u	$⊢ U = DVecH (K) (W)$
3		hdmap1l6.v	$⊢ V = {Base}_{U}$
4		hdmap1l6.p	$⊢ \underset{˙}{+} = +_{U}$
5		hdmap1l6.s	$⊢ \underset{˙}{-} = -_{U}$
6		hdmap1l6c.o	$⊢ \underset{˙}{0} = 0_{U}$
7		hdmap1l6.n	$⊢ N = LSpan (U)$
8		hdmap1l6.c	$⊢ C = LCDual (K) (W)$
9		hdmap1l6.d	$⊢ D = {Base}_{C}$
10		hdmap1l6.a	$⊢ \underset{˙}{✚} = +_{C}$
11		hdmap1l6.r	$⊢ R = -_{C}$
12		hdmap1l6.q	$⊢ Q = 0_{C}$
13		hdmap1l6.l	$⊢ L = LSpan (C)$
14		hdmap1l6.m	$⊢ M = mapd (K) (W)$
15		hdmap1l6.i	$⊢ I = HDMap1 (K) (W)$
16		hdmap1l6.k	$⊢ φ \to (K \in HL \land W \in H)$
17		hdmap1l6.f	$⊢ φ \to F \in D$
18		hdmap1l6cl.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
19		hdmap1l6.mn	$⊢ φ \to M (N (\{X\})) = L (\{F\})$
20		hdmap1l6e.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
21		hdmap1l6e.z	$⊢ φ \to Z \in (V ∖ \{\underset{˙}{0}\})$
22		hdmap1l6e.xn	$⊢ φ \to \neg X \in N (\{Y, Z\})$
23		hdmap1l6.yz	$⊢ φ \to N (\{Y\}) \neq N (\{Z\})$
24		hdmap1l6.fg	$⊢ φ \to I (⟨X, F, Y⟩) = G$
25		hdmap1l6.fe	$⊢ φ \to I (⟨X, F, Z⟩) = E$
26		eqid	$⊢ LSubSp (U) = LSubSp (U)$
27	1 2 16	dvhlmod	$⊢ φ \to U \in LMod$
28	18	eldifad	$⊢ φ \to X \in V$
29	20	eldifad	$⊢ φ \to Y \in V$
30	3 5	lmodvsubcl	$⊢ (U \in LMod \land X \in V \land Y \in V) \to X \underset{˙}{-} Y \in V$
31	27 28 29 30	syl3anc	$⊢ φ \to X \underset{˙}{-} Y \in V$
32	3 26 7	lspsncl	$⊢ (U \in LMod \land X \underset{˙}{-} Y \in V) \to N (\{X \underset{˙}{-} Y\}) \in LSubSp (U)$
33	27 31 32	syl2anc	$⊢ φ \to N (\{X \underset{˙}{-} Y\}) \in LSubSp (U)$
34	21	eldifad	$⊢ φ \to Z \in V$
35	3 26 7	lspsncl	$⊢ (U \in LMod \land Z \in V) \to N (\{Z\}) \in LSubSp (U)$
36	27 34 35	syl2anc	$⊢ φ \to N (\{Z\}) \in LSubSp (U)$
37		eqid	$⊢ LSSum (U) = LSSum (U)$
38	26 37	lsmcl	$⊢ (U \in LMod \land N (\{X \underset{˙}{-} Y\}) \in LSubSp (U) \land N (\{Z\}) \in LSubSp (U)) \to N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\}) \in LSubSp (U)$
39	27 33 36 38	syl3anc	$⊢ φ \to N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\}) \in LSubSp (U)$
40	3 5	lmodvsubcl	$⊢ (U \in LMod \land X \in V \land Z \in V) \to X \underset{˙}{-} Z \in V$
41	27 28 34 40	syl3anc	$⊢ φ \to X \underset{˙}{-} Z \in V$
42	3 26 7	lspsncl	$⊢ (U \in LMod \land X \underset{˙}{-} Z \in V) \to N (\{X \underset{˙}{-} Z\}) \in LSubSp (U)$
43	27 41 42	syl2anc	$⊢ φ \to N (\{X \underset{˙}{-} Z\}) \in LSubSp (U)$
44	3 26 7	lspsncl	$⊢ (U \in LMod \land Y \in V) \to N (\{Y\}) \in LSubSp (U)$
45	27 29 44	syl2anc	$⊢ φ \to N (\{Y\}) \in LSubSp (U)$
46	26 37	lsmcl	$⊢ (U \in LMod \land N (\{X \underset{˙}{-} Z\}) \in LSubSp (U) \land N (\{Y\}) \in LSubSp (U)) \to N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\}) \in LSubSp (U)$
47	27 43 45 46	syl3anc	$⊢ φ \to N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\}) \in LSubSp (U)$
48	1 14 2 26 16 39 47	mapdin	$⊢ φ \to M ((N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\})) \cap (N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\}))) = M (N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\})) \cap M (N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\}))$
49		eqid	$⊢ LSSum (C) = LSSum (C)$
50	1 14 2 26 37 8 49 16 33 36	mapdlsm	$⊢ φ \to M (N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\})) = M (N (\{X \underset{˙}{-} Y\})) LSSum (C) M (N (\{Z\}))$
51	1 14 2 26 37 8 49 16 43 45	mapdlsm	$⊢ φ \to M (N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\})) = M (N (\{X \underset{˙}{-} Z\})) LSSum (C) M (N (\{Y\}))$
52	50 51	ineq12d	$⊢ φ \to M (N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\})) \cap M (N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\})) = (M (N (\{X \underset{˙}{-} Y\})) LSSum (C) M (N (\{Z\}))) \cap (M (N (\{X \underset{˙}{-} Z\})) LSSum (C) M (N (\{Y\})))$
53	1 2 16	dvhlvec	$⊢ φ \to U \in LVec$
54	3 6 7 53 29 21 28 23 22	lspindp2	$⊢ φ \to (N (\{X\}) \neq N (\{Y\}) \land \neg Z \in N (\{X, Y\}))$
55	54	simpld	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
56	1 2 3 6 7 8 9 13 14 15 16 17 19 55 18 29	hdmap1cl	$⊢ φ \to I (⟨X, F, Y⟩) \in D$
57	24 56	eqeltrrd	$⊢ φ \to G \in D$
58	1 2 3 5 6 7 8 9 11 13 14 15 16 18 17 20 57 55 19	hdmap1eq	$⊢ φ \to (I (⟨X, F, Y⟩) = G \leftrightarrow (M (N (\{Y\})) = L (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = L (\{F R G\})))$
59	24 58	mpbid	$⊢ φ \to (M (N (\{Y\})) = L (\{G\}) \land M (N (\{X \underset{˙}{-} Y\})) = L (\{F R G\}))$
60	59	simprd	$⊢ φ \to M (N (\{X \underset{˙}{-} Y\})) = L (\{F R G\})$
61	3 6 7 53 20 34 28 23 22	lspindp1	$⊢ φ \to (N (\{X\}) \neq N (\{Z\}) \land \neg Y \in N (\{X, Z\}))$
62	61	simpld	$⊢ φ \to N (\{X\}) \neq N (\{Z\})$
63	1 2 3 6 7 8 9 13 14 15 16 17 19 62 18 34	hdmap1cl	$⊢ φ \to I (⟨X, F, Z⟩) \in D$
64	25 63	eqeltrrd	$⊢ φ \to E \in D$
65	1 2 3 5 6 7 8 9 11 13 14 15 16 18 17 21 64 62 19	hdmap1eq	$⊢ φ \to (I (⟨X, F, Z⟩) = E \leftrightarrow (M (N (\{Z\})) = L (\{E\}) \land M (N (\{X \underset{˙}{-} Z\})) = L (\{F R E\})))$
66	25 65	mpbid	$⊢ φ \to (M (N (\{Z\})) = L (\{E\}) \land M (N (\{X \underset{˙}{-} Z\})) = L (\{F R E\}))$
67	66	simpld	$⊢ φ \to M (N (\{Z\})) = L (\{E\})$
68	60 67	oveq12d	$⊢ φ \to M (N (\{X \underset{˙}{-} Y\})) LSSum (C) M (N (\{Z\})) = L (\{F R G\}) LSSum (C) L (\{E\})$
69	66	simprd	$⊢ φ \to M (N (\{X \underset{˙}{-} Z\})) = L (\{F R E\})$
70	59	simpld	$⊢ φ \to M (N (\{Y\})) = L (\{G\})$
71	69 70	oveq12d	$⊢ φ \to M (N (\{X \underset{˙}{-} Z\})) LSSum (C) M (N (\{Y\})) = L (\{F R E\}) LSSum (C) L (\{G\})$
72	68 71	ineq12d	$⊢ φ \to (M (N (\{X \underset{˙}{-} Y\})) LSSum (C) M (N (\{Z\}))) \cap (M (N (\{X \underset{˙}{-} Z\})) LSSum (C) M (N (\{Y\}))) = (L (\{F R G\}) LSSum (C) L (\{E\})) \cap (L (\{F R E\}) LSSum (C) L (\{G\}))$
73	52 72	eqtrd	$⊢ φ \to M (N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\})) \cap M (N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\})) = (L (\{F R G\}) LSSum (C) L (\{E\})) \cap (L (\{F R E\}) LSSum (C) L (\{G\}))$
74	48 73	eqtrd	$⊢ φ \to M ((N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\})) \cap (N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\}))) = (L (\{F R G\}) LSSum (C) L (\{E\})) \cap (L (\{F R E\}) LSSum (C) L (\{G\}))$
75	3 5 6 37 7 53 28 22 23 20 21 4	baerlem5a	$⊢ φ \to N (\{X \underset{˙}{-} (Y \underset{˙}{+} Z)\}) = (N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\})) \cap (N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\}))$
76	75	fveq2d	$⊢ φ \to M (N (\{X \underset{˙}{-} (Y \underset{˙}{+} Z)\})) = M ((N (\{X \underset{˙}{-} Y\}) LSSum (U) N (\{Z\})) \cap (N (\{X \underset{˙}{-} Z\}) LSSum (U) N (\{Y\})))$
77	1 8 16	lcdlvec	$⊢ φ \to C \in LVec$
78	1 14 2 3 7 8 9 13 16 17 19 28 29 57 70 34 64 67 22	mapdindp	$⊢ φ \to \neg F \in L (\{G, E\})$
79	1 14 2 3 7 8 9 13 16 57 70 29 34 64 67 23	mapdncol	$⊢ φ \to L (\{G\}) \neq L (\{E\})$
80	1 14 2 3 7 8 9 13 16 57 70 6 12 20	mapdn0	$⊢ φ \to G \in (D ∖ \{Q\})$
81	1 14 2 3 7 8 9 13 16 64 67 6 12 21	mapdn0	$⊢ φ \to E \in (D ∖ \{Q\})$
82	9 11 12 49 13 77 17 78 79 80 81 10	baerlem5a	$⊢ φ \to L (\{F R (G \underset{˙}{✚} E)\}) = (L (\{F R G\}) LSSum (C) L (\{E\})) \cap (L (\{F R E\}) LSSum (C) L (\{G\}))$
83	74 76 82	3eqtr4d	$⊢ φ \to M (N (\{X \underset{˙}{-} (Y \underset{˙}{+} Z)\})) = L (\{F R (G \underset{˙}{✚} E)\})$

Metamath Proof Explorer

Theorem hdmap1l6lem1

Proof