cdleme32c

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	cdleme32c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to F (X) \underset{˙}{\leq} F (Y)$

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		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to X \underset{˙}{\leq} Y$
16		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to K \in HL$
17	16	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to K \in Lat$
18		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to X \in B$
19		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to Y \in B$
20		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to W \in H$
21	1 6	lhpbase	$⊢ W \in H \to W \in B$
22	20 21	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to W \in B$
23	1 2 4	latmlem1	$⊢ (K \in Lat \land (X \in B \land Y \in B \land W \in B)) \to (X \underset{˙}{\leq} Y \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W))$
24	17 18 19 22 23	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\leq} Y \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W))$
25	15 24	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W)$
26	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
27	17 18 22 26	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to X \underset{˙}{\land} W \in B$
28	1 4	latmcl	$⊢ (K \in Lat \land Y \in B \land W \in B) \to Y \underset{˙}{\land} W \in B$
29	17 19 22 28	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to Y \underset{˙}{\land} W \in B$
30		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
31		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
32		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
33		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to P \neq Q$
34	1 2 3 4 5 6 7 8 9 10 11 12	cdleme27cl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land P \neq Q)) \to N \in B$
35	16 20 30 31 32 33 34	syl222anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to N \in B$
36	1 2 3	latjlej2	$⊢ (K \in Lat \land (X \underset{˙}{\land} W \in B \land Y \underset{˙}{\land} W \in B \land N \in B)) \to ((X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W) \to (N \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)))$
37	17 27 29 35 36	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to ((X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W) \to (N \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)))$
38	25 37	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (N \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
39		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
40		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to (P \neq Q \land \neg X \underset{˙}{\leq} W)$
41		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
42	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) = N \underset{˙}{\lor} (X \underset{˙}{\land} W)$
43	39 18 40 32 41 42	syl122anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to F (X) = N \underset{˙}{\lor} (X \underset{˙}{\land} W)$
44	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to F (Y) = N \underset{˙}{\lor} (Y \underset{˙}{\land} W)$
45	38 43 44	3brtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land X \underset{˙}{\leq} Y)) \to F (X) \underset{˙}{\leq} F (Y)$

Metamath Proof Explorer

Theorem cdleme32c

Proof