dihord11b

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

	Ref	Expression
Hypotheses	dihjust.b	$⊢ B = {Base}_{K}$
	dihjust.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	dihjust.j	$⊢ \underset{˙}{\lor} = join (K)$
	dihjust.m	$⊢ \underset{˙}{\land} = meet (K)$
	dihjust.a	$⊢ A = Atoms (K)$
	dihjust.h	$⊢ H = LHyp (K)$
	dihjust.i	$⊢ I = DIsoB (K) (W)$
	dihjust.J	$⊢ J = DIsoC (K) (W)$
	dihjust.u	$⊢ U = DVecH (K) (W)$
	dihjust.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
	dihord2c.t	$⊢ T = LTrn (K) (W)$
	dihord2c.r	$⊢ R = trL (K) (W)$
	dihord2c.o	$⊢ O = (h \in T ⟼ {I ↾}_{B})$
	dihord2.p	$⊢ P = oc (K) (W)$
	dihord2.e	$⊢ E = TEndo (K) (W)$
	dihord2.d	$⊢ \underset{˙}{+} = +_{U}$
	dihord2.g	$⊢ G = (ι h \in T \| h (P) = N)$
Assertion	dihord11b	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to ⟨f, O⟩ \in (J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))$

Proof

Step	Hyp	Ref	Expression
1		dihjust.b	$⊢ B = {Base}_{K}$
2		dihjust.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		dihjust.j	$⊢ \underset{˙}{\lor} = join (K)$
4		dihjust.m	$⊢ \underset{˙}{\land} = meet (K)$
5		dihjust.a	$⊢ A = Atoms (K)$
6		dihjust.h	$⊢ H = LHyp (K)$
7		dihjust.i	$⊢ I = DIsoB (K) (W)$
8		dihjust.J	$⊢ J = DIsoC (K) (W)$
9		dihjust.u	$⊢ U = DVecH (K) (W)$
10		dihjust.s	$⊢ \underset{˙}{\oplus} = LSSum (U)$
11		dihord2c.t	$⊢ T = LTrn (K) (W)$
12		dihord2c.r	$⊢ R = trL (K) (W)$
13		dihord2c.o	$⊢ O = (h \in T ⟼ {I ↾}_{B})$
14		dihord2.p	$⊢ P = oc (K) (W)$
15		dihord2.e	$⊢ E = TEndo (K) (W)$
16		dihord2.d	$⊢ \underset{˙}{+} = +_{U}$
17		dihord2.g	$⊢ G = (ι h \in T \| h (P) = N)$
18	1 2 3 4 5 6 7 8 9 10	dihord2b	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \to I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$
19	18	adantr	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)$
20		simpr	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))$
21		eqidd	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to O = O$
22		simpl11	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to (K \in HL \land W \in H)$
23		simp11l	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \to K \in HL$
24	23	adantr	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to K \in HL$
25	24	hllatd	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to K \in Lat$
26		simpl2l	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to X \in B$
27		simp11r	$⊢ (((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \to W \in H$
28	27	adantr	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to W \in H$
29	1 6	lhpbase	$⊢ W \in H \to W \in B$
30	28 29	syl	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to W \in B$
31	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
32	25 26 30 31	syl3anc	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to X \underset{˙}{\land} W \in B$
33	1 2 4	latmle2	$⊢ (K \in Lat \land X \in B \land W \in B) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} W$
34	25 26 30 33	syl3anc	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} W$
35	1 2 6 11 12 13 7	dibopelval3	$⊢ ((K \in HL \land W \in H) \land (X \underset{˙}{\land} W \in B \land (X \underset{˙}{\land} W) \underset{˙}{\leq} W)) \to (⟨f, O⟩ \in I (X \underset{˙}{\land} W) \leftrightarrow ((f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W)) \land O = O))$
36	22 32 34 35	syl12anc	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to (⟨f, O⟩ \in I (X \underset{˙}{\land} W) \leftrightarrow ((f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W)) \land O = O))$
37	20 21 36	mpbir2and	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to ⟨f, O⟩ \in I (X \underset{˙}{\land} W)$
38	19 37	sseldd	$⊢ ((((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (N \in A \land \neg N \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land J (Q) \underset{˙}{\oplus} I (X \underset{˙}{\land} W) \subseteq J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W)) \land (f \in T \land R (f) \underset{˙}{\leq} (X \underset{˙}{\land} W))) \to ⟨f, O⟩ \in (J (N) \underset{˙}{\oplus} I (Y \underset{˙}{\land} W))$

Metamath Proof Explorer

Theorem dihord11b

Proof