dihordlem7

Description: Part of proof of Lemma N of Crawley p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014)

	Ref	Expression
Hypotheses	dihordlem8.b	$⊢ B = {Base}_{K}$
	dihordlem8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	dihordlem8.a	$⊢ A = Atoms (K)$
	dihordlem8.h	$⊢ H = LHyp (K)$
	dihordlem8.p	$⊢ P = oc (K) (W)$
	dihordlem8.o	$⊢ O = (h \in T ⟼ {I ↾}_{B})$
	dihordlem8.t	$⊢ T = LTrn (K) (W)$
	dihordlem8.e	$⊢ E = TEndo (K) (W)$
	dihordlem8.u	$⊢ U = DVecH (K) (W)$
	dihordlem8.s	$⊢ \underset{˙}{+} = +_{U}$
	dihordlem8.g	$⊢ G = (ι h \in T \| h (P) = R)$
Assertion	dihordlem7	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (f = s (G) \circ g \land O = s)$

Proof

Step	Hyp	Ref	Expression
1		dihordlem8.b	$⊢ B = {Base}_{K}$
2		dihordlem8.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dihordlem8.a	$⊢ A = Atoms (K)$
4		dihordlem8.h	$⊢ H = LHyp (K)$
5		dihordlem8.p	$⊢ P = oc (K) (W)$
6		dihordlem8.o	$⊢ O = (h \in T ⟼ {I ↾}_{B})$
7		dihordlem8.t	$⊢ T = LTrn (K) (W)$
8		dihordlem8.e	$⊢ E = TEndo (K) (W)$
9		dihordlem8.u	$⊢ U = DVecH (K) (W)$
10		dihordlem8.s	$⊢ \underset{˙}{+} = +_{U}$
11		dihordlem8.g	$⊢ G = (ι h \in T \| h (P) = R)$
12		simp33	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩$
13		simp1	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (K \in HL \land W \in H)$
14		simp2l	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
15		simp2r	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
16		simp31	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to s \in E$
17		simp32	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to g \in T$
18	1 2 3 4 5 6 7 8 9 10 11	dihordlem6	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T)) \to ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩ = ⟨s (G) \circ g, s⟩$
19	13 14 15 16 17 18	syl122anc	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩ = ⟨s (G) \circ g, s⟩$
20	12 19	eqtrd	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to ⟨f, O⟩ = ⟨s (G) \circ g, s⟩$
21		fvex	$⊢ s (G) \in V$
22		vex	$⊢ g \in V$
23	21 22	coex	$⊢ s (G) \circ g \in V$
24		vex	$⊢ s \in V$
25	23 24	opth2	$⊢ ⟨f, O⟩ = ⟨s (G) \circ g, s⟩ \leftrightarrow (f = s (G) \circ g \land O = s)$
26	20 25	sylib	$⊢ ((K \in HL \land W \in H) \land ((Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (s \in E \land g \in T \land ⟨f, O⟩ = ⟨s (G), s⟩ \underset{˙}{+} ⟨g, O⟩)) \to (f = s (G) \circ g \land O = s)$

Metamath Proof Explorer

Theorem dihordlem7

Proof