hdmap1eulemOLDN

Description: Lemma for hdmap1euOLDN . TODO: combine with hdmap1euOLDN or at least share some hypotheses. (Contributed by NM, 15-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hdmap1eulem.h	$⊢ H = LHyp (K)$
	hdmap1eulem.u	$⊢ U = DVecH (K) (W)$
	hdmap1eulem.v	$⊢ V = {Base}_{U}$
	hdmap1eulem.s	$⊢ \underset{˙}{-} = -_{U}$
	hdmap1eulem.o	$⊢ \underset{˙}{0} = 0_{U}$
	hdmap1eulem.n	$⊢ N = LSpan (U)$
	hdmap1eulem.c	$⊢ C = LCDual (K) (W)$
	hdmap1eulem.d	$⊢ D = {Base}_{C}$
	hdmap1eulem.r	$⊢ R = -_{C}$
	hdmap1eulem.q	$⊢ Q = 0_{C}$
	hdmap1eulem.j	$⊢ J = LSpan (C)$
	hdmap1eulem.m	$⊢ M = mapd (K) (W)$
	hdmap1eulem.i	$⊢ I = HDMap1 (K) (W)$
	hdmap1eulem.k	$⊢ φ \to (K \in HL \land W \in H)$
	hdmap1eulem.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
	hdmap1eulem.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
	hdmap1eulem.f	$⊢ φ \to F \in D$
	hdmap1eulem.y	$⊢ φ \to T \in V$
	hdmap1eulem.l	$⊢ L = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
Assertion	hdmap1eulemOLDN	$⊢ φ \to \exists! y \in D \forall z \in V (\neg z \in N (\{X, T\}) \to y = I (⟨z, I (⟨X, F, z⟩), T⟩))$

Proof

Step	Hyp	Ref	Expression
1		hdmap1eulem.h	$⊢ H = LHyp (K)$
2		hdmap1eulem.u	$⊢ U = DVecH (K) (W)$
3		hdmap1eulem.v	$⊢ V = {Base}_{U}$
4		hdmap1eulem.s	$⊢ \underset{˙}{-} = -_{U}$
5		hdmap1eulem.o	$⊢ \underset{˙}{0} = 0_{U}$
6		hdmap1eulem.n	$⊢ N = LSpan (U)$
7		hdmap1eulem.c	$⊢ C = LCDual (K) (W)$
8		hdmap1eulem.d	$⊢ D = {Base}_{C}$
9		hdmap1eulem.r	$⊢ R = -_{C}$
10		hdmap1eulem.q	$⊢ Q = 0_{C}$
11		hdmap1eulem.j	$⊢ J = LSpan (C)$
12		hdmap1eulem.m	$⊢ M = mapd (K) (W)$
13		hdmap1eulem.i	$⊢ I = HDMap1 (K) (W)$
14		hdmap1eulem.k	$⊢ φ \to (K \in HL \land W \in H)$
15		hdmap1eulem.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
16		hdmap1eulem.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
17		hdmap1eulem.f	$⊢ φ \to F \in D$
18		hdmap1eulem.y	$⊢ φ \to T \in V$
19		hdmap1eulem.l	$⊢ L = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
20	1 2 3 4 5 6 7 8 9 10 11 12 19 14 17 15 16 18	mapdh9aOLDN	$⊢ φ \to \exists! y \in D \forall z \in V (\neg z \in N (\{X, T\}) \to y = L (⟨z, L (⟨X, F, z⟩), T⟩))$
21	14	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to (K \in HL \land W \in H)$
22	16	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to X \in (V ∖ \{\underset{˙}{0}\})$
23	17	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to F \in D$
24		simplr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to z \in V$
25	1 2 3 4 5 6 7 8 9 10 11 12 13 21 22 23 24 19	hdmap1valc	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to I (⟨X, F, z⟩) = L (⟨X, F, z⟩)$
26	25	oteq2d	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to ⟨z, I (⟨X, F, z⟩), T⟩ = ⟨z, L (⟨X, F, z⟩), T⟩$
27	26	fveq2d	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to I (⟨z, I (⟨X, F, z⟩), T⟩) = I (⟨z, L (⟨X, F, z⟩), T⟩)$
28		eqid	$⊢ LSubSp (U) = LSubSp (U)$
29	1 2 14	dvhlmod	$⊢ φ \to U \in LMod$
30	29	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to U \in LMod$
31	16	eldifad	$⊢ φ \to X \in V$
32	3 28 6 29 31 18	lspprcl	$⊢ φ \to N (\{X, T\}) \in LSubSp (U)$
33	32	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to N (\{X, T\}) \in LSubSp (U)$
34		simpr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to \neg z \in N (\{X, T\})$
35	5 28 30 33 24 34	lssneln0	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to z \in (V ∖ \{\underset{˙}{0}\})$
36	15	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to M (N (\{X\})) = J (\{F\})$
37	1 2 14	dvhlvec	$⊢ φ \to U \in LVec$
38	37	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to U \in LVec$
39	31	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to X \in V$
40	18	ad2antrr	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to T \in V$
41	3 6 38 24 39 40 34	lspindpi	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to (N (\{z\}) \neq N (\{X\}) \land N (\{z\}) \neq N (\{T\}))$
42	41	simpld	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to N (\{z\}) \neq N (\{X\})$
43	42	necomd	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to N (\{X\}) \neq N (\{z\})$
44	10 19 1 12 2 3 4 5 6 7 8 9 11 21 23 36 22 24 43	mapdhcl	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to L (⟨X, F, z⟩) \in D$
45	1 2 3 4 5 6 7 8 9 10 11 12 13 21 35 44 40 19	hdmap1valc	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to I (⟨z, L (⟨X, F, z⟩), T⟩) = L (⟨z, L (⟨X, F, z⟩), T⟩)$
46	27 45	eqtrd	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to I (⟨z, I (⟨X, F, z⟩), T⟩) = L (⟨z, L (⟨X, F, z⟩), T⟩)$
47	46	eqeq2d	$⊢ ((φ \land z \in V) \land \neg z \in N (\{X, T\})) \to (y = I (⟨z, I (⟨X, F, z⟩), T⟩) \leftrightarrow y = L (⟨z, L (⟨X, F, z⟩), T⟩))$
48	47	pm5.74da	$⊢ (φ \land z \in V) \to ((\neg z \in N (\{X, T\}) \to y = I (⟨z, I (⟨X, F, z⟩), T⟩)) \leftrightarrow (\neg z \in N (\{X, T\}) \to y = L (⟨z, L (⟨X, F, z⟩), T⟩)))$
49	48	ralbidva	$⊢ φ \to (\forall z \in V (\neg z \in N (\{X, T\}) \to y = I (⟨z, I (⟨X, F, z⟩), T⟩)) \leftrightarrow \forall z \in V (\neg z \in N (\{X, T\}) \to y = L (⟨z, L (⟨X, F, z⟩), T⟩)))$
50	49	reubidv	$⊢ φ \to (\exists! y \in D \forall z \in V (\neg z \in N (\{X, T\}) \to y = I (⟨z, I (⟨X, F, z⟩), T⟩)) \leftrightarrow \exists! y \in D \forall z \in V (\neg z \in N (\{X, T\}) \to y = L (⟨z, L (⟨X, F, z⟩), T⟩)))$
51	20 50	mpbird	$⊢ φ \to \exists! y \in D \forall z \in V (\neg z \in N (\{X, T\}) \to y = I (⟨z, I (⟨X, F, z⟩), T⟩))$

Metamath Proof Explorer

Theorem hdmap1eulemOLDN

Proof