cdleme32b

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 19-Feb-2013)

	Ref	Expression
Hypotheses	cdleme32.b	$⊢ B = {Base}_{K}$
	cdleme32.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme32.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme32.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme32.a	$⊢ A = Atoms (K)$
	cdleme32.h	$⊢ H = LHyp (K)$
	cdleme32.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme32.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme32.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme32.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme32.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
	cdleme32.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
	cdleme32.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
	cdleme32.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
Assertion	cdleme32b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to F (Y) = N \underset{˙}{\lor} (Y \underset{˙}{\land} W)$

Proof

Step	Hyp	Ref	Expression
1		cdleme32.b	$⊢ B = {Base}_{K}$
2		cdleme32.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme32.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme32.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme32.a	$⊢ A = Atoms (K)$
6		cdleme32.h	$⊢ H = LHyp (K)$
7		cdleme32.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme32.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme32.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdleme32.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
11		cdleme32.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
12		cdleme32.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
13		cdleme32.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
14		cdleme32.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
15		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
16		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to Y \in B$
17		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to P \neq Q$
18		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to \neg X \underset{˙}{\leq} W$
19		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to X \underset{˙}{\leq} Y$
20		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to K \in HL$
21	20	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to K \in Lat$
22		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to X \in B$
23		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to W \in H$
24	1 6	lhpbase	$⊢ W \in H \to W \in B$
25	23 24	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to W \in B$
26	1 2	lattr	$⊢ (K \in Lat \land (X \in B \land Y \in B \land W \in B)) \to ((X \underset{˙}{\leq} Y \land Y \underset{˙}{\leq} W) \to X \underset{˙}{\leq} W)$
27	21 22 16 25 26	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to ((X \underset{˙}{\leq} Y \land Y \underset{˙}{\leq} W) \to X \underset{˙}{\leq} W)$
28	19 27	mpand	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (Y \underset{˙}{\leq} W \to X \underset{˙}{\leq} W)$
29	18 28	mtod	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to \neg Y \underset{˙}{\leq} W$
30	17 29	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (P \neq Q \land \neg Y \underset{˙}{\leq} W)$
31		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
32		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (K \in HL \land W \in H)$
33		simp31l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to s \in A$
34	22 18	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
35		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
36	1 2 3 4 5 6	cdleme30a	$⊢ ((K \in HL \land W \in H) \land (s \in A \land (X \in B \land \neg X \underset{˙}{\leq} W) \land Y \in B) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y$
37	32 33 34 16 35 19 36	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y$
38	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (Y \in B \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y)) \to F (Y) = N \underset{˙}{\lor} (Y \underset{˙}{\land} W)$
39	15 16 30 31 37 38	syl122anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to F (Y) = N \underset{˙}{\lor} (Y \underset{˙}{\land} W)$

Metamath Proof Explorer

Theorem cdleme32b

Proof