mapdh8g

Description: Part of Part (8) in Baer p. 48. Eliminate X e. ( N{ Y , T } ) . TODO: break out T =/= .0. in mapdh8e so we can share hypotheses. Also, look at hypothesis sharing for earlier mapdh8* and mapdh75* stuff. (Contributed by NM, 10-May-2015)

	Ref	Expression
Hypotheses	mapdh8a.h	$⊢ H = LHyp (K)$
	mapdh8a.u	$⊢ U = DVecH (K) (W)$
	mapdh8a.v	$⊢ V = {Base}_{U}$
	mapdh8a.s	$⊢ \underset{˙}{-} = -_{U}$
	mapdh8a.o	$⊢ \underset{˙}{0} = 0_{U}$
	mapdh8a.n	$⊢ N = LSpan (U)$
	mapdh8a.c	$⊢ C = LCDual (K) (W)$
	mapdh8a.d	$⊢ D = {Base}_{C}$
	mapdh8a.r	$⊢ R = -_{C}$
	mapdh8a.q	$⊢ Q = 0_{C}$
	mapdh8a.j	$⊢ J = LSpan (C)$
	mapdh8a.m	$⊢ M = mapd (K) (W)$
	mapdh8a.i	$⊢ I = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
	mapdh8a.k	$⊢ φ \to (K \in HL \land W \in H)$
	mapdh8e.f	$⊢ φ \to F \in D$
	mapdh8e.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
	mapdh8e.eg	$⊢ φ \to I (⟨X, F, Y⟩) = G$
	mapdh8e.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8e.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8e.t	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
	mapdh8e.xy	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
	mapdh8e.xt	$⊢ φ \to N (\{X\}) \neq N (\{T\})$
	mapdh8e.yt	$⊢ φ \to N (\{Y\}) \neq N (\{T\})$
Assertion	mapdh8g	$⊢ φ \to I (⟨Y, G, T⟩) = I (⟨X, F, T⟩)$

Proof

Step	Hyp	Ref	Expression
1		mapdh8a.h	$⊢ H = LHyp (K)$
2		mapdh8a.u	$⊢ U = DVecH (K) (W)$
3		mapdh8a.v	$⊢ V = {Base}_{U}$
4		mapdh8a.s	$⊢ \underset{˙}{-} = -_{U}$
5		mapdh8a.o	$⊢ \underset{˙}{0} = 0_{U}$
6		mapdh8a.n	$⊢ N = LSpan (U)$
7		mapdh8a.c	$⊢ C = LCDual (K) (W)$
8		mapdh8a.d	$⊢ D = {Base}_{C}$
9		mapdh8a.r	$⊢ R = -_{C}$
10		mapdh8a.q	$⊢ Q = 0_{C}$
11		mapdh8a.j	$⊢ J = LSpan (C)$
12		mapdh8a.m	$⊢ M = mapd (K) (W)$
13		mapdh8a.i	$⊢ I = (x \in V ⟼ if (2^{nd} (x) = \underset{˙}{0}, Q, (ι h \in D \| (M (N (\{2^{nd} (x)\})) = J (\{h\}) \land M (N (\{1^{st} (1^{st} (x)) \underset{˙}{-} 2^{nd} (x)\})) = J (\{2^{nd} (1^{st} (x)) R h\})))))$
14		mapdh8a.k	$⊢ φ \to (K \in HL \land W \in H)$
15		mapdh8e.f	$⊢ φ \to F \in D$
16		mapdh8e.mn	$⊢ φ \to M (N (\{X\})) = J (\{F\})$
17		mapdh8e.eg	$⊢ φ \to I (⟨X, F, Y⟩) = G$
18		mapdh8e.x	$⊢ φ \to X \in (V ∖ \{\underset{˙}{0}\})$
19		mapdh8e.y	$⊢ φ \to Y \in (V ∖ \{\underset{˙}{0}\})$
20		mapdh8e.t	$⊢ φ \to T \in (V ∖ \{\underset{˙}{0}\})$
21		mapdh8e.xy	$⊢ φ \to N (\{X\}) \neq N (\{Y\})$
22		mapdh8e.xt	$⊢ φ \to N (\{X\}) \neq N (\{T\})$
23		mapdh8e.yt	$⊢ φ \to N (\{Y\}) \neq N (\{T\})$
24	14	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to (K \in HL \land W \in H)$
25	15	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to F \in D$
26	16	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to M (N (\{X\})) = J (\{F\})$
27	17	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to I (⟨X, F, Y⟩) = G$
28	18	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to X \in (V ∖ \{\underset{˙}{0}\})$
29	19	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to Y \in (V ∖ \{\underset{˙}{0}\})$
30	20	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to T \in (V ∖ \{\underset{˙}{0}\})$
31	21	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to N (\{X\}) \neq N (\{Y\})$
32	22	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to N (\{X\}) \neq N (\{T\})$
33	23	adantr	$⊢ (φ \land X \in N (\{Y, T\})) \to N (\{Y\}) \neq N (\{T\})$
34		simpr	$⊢ (φ \land X \in N (\{Y, T\})) \to X \in N (\{Y, T\})$
35	1 2 3 4 5 6 7 8 9 10 11 12 13 24 25 26 27 28 29 30 31 32 33 34	mapdh8e	$⊢ (φ \land X \in N (\{Y, T\})) \to I (⟨Y, G, T⟩) = I (⟨X, F, T⟩)$
36	14	adantr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to (K \in HL \land W \in H)$
37	15	adantr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to F \in D$
38	16	adantr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to M (N (\{X\})) = J (\{F\})$
39	17	adantr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to I (⟨X, F, Y⟩) = G$
40	18	adantr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to X \in (V ∖ \{\underset{˙}{0}\})$
41	19	adantr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to Y \in (V ∖ \{\underset{˙}{0}\})$
42	23	adantr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to N (\{Y\}) \neq N (\{T\})$
43	20	adantr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to T \in (V ∖ \{\underset{˙}{0}\})$
44		simpr	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to \neg X \in N (\{Y, T\})$
45	1 2 3 4 5 6 7 8 9 10 11 12 13 36 37 38 39 40 41 42 43 44	mapdh8a	$⊢ (φ \land \neg X \in N (\{Y, T\})) \to I (⟨Y, G, T⟩) = I (⟨X, F, T⟩)$
46	35 45	pm2.61dan	$⊢ φ \to I (⟨Y, G, T⟩) = I (⟨X, F, T⟩)$

Metamath Proof Explorer

Theorem mapdh8g

Proof