cdleme32e

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 20-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	cdleme32e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to F (X) \underset{˙}{\leq} F (Y)$

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		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to P \neq Q$
16	15	pm2.24d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (\neg P \neq Q \to X \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)))$
17		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to K \in HL$
18	17	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to K \in Lat$
19		simp21l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to X \in B$
20		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to W \in H$
21	1 6	lhpbase	$⊢ W \in H \to W \in B$
22	20 21	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to W \in B$
23	1 2 4	latleeqm1	$⊢ (K \in Lat \land X \in B \land W \in B) \to (X \underset{˙}{\leq} W \leftrightarrow X \underset{˙}{\land} W = X)$
24	18 19 22 23	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\leq} W \leftrightarrow X \underset{˙}{\land} W = X)$
25	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
26	18 19 22 25	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to X \underset{˙}{\land} W \in B$
27		simp21r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Y \in B$
28	1 4	latmcl	$⊢ (K \in Lat \land Y \in B \land W \in B) \to Y \underset{˙}{\land} W \in B$
29	18 27 22 28	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to Y \underset{˙}{\land} W \in B$
30		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (K \in HL \land W \in H)$
31		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
32		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
33		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
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	30 31 32 33 15 34	syl122anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to N \in B$
36	1 3	latjcl	$⊢ (K \in Lat \land N \in B \land Y \underset{˙}{\land} W \in B) \to N \underset{˙}{\lor} (Y \underset{˙}{\land} W) \in B$
37	18 35 29 36	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to N \underset{˙}{\lor} (Y \underset{˙}{\land} W) \in B$
38		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to X \underset{˙}{\leq} Y$
39	1 2 4	latmlem1	$⊢ (K \in Lat \land (X \in B \land Y \in B \land W \in B)) \to (X \underset{˙}{\leq} Y \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W))$
40	18 19 27 22 39	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\leq} Y \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W))$
41	38 40	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (Y \underset{˙}{\land} W)$
42	1 2 3	latlej2	$⊢ (K \in Lat \land N \in B \land Y \underset{˙}{\land} W \in B) \to (Y \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
43	18 35 29 42	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (Y \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
44	1 2 18 26 29 37 41 43	lattrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
45		breq1	$⊢ X \underset{˙}{\land} W = X \to ((X \underset{˙}{\land} W) \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)) \leftrightarrow X \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)))$
46	44 45	syl5ibcom	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\land} W = X \to X \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)))$
47	24 46	sylbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (X \underset{˙}{\leq} W \to X \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W)))$
48		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)$
49		pm4.53	$⊢ \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \leftrightarrow (\neg P \neq Q \lor X \underset{˙}{\leq} W)$
50	48 49	sylib	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (\neg P \neq Q \lor X \underset{˙}{\leq} W)$
51	16 47 50	mpjaod	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to X \underset{˙}{\leq} (N \underset{˙}{\lor} (Y \underset{˙}{\land} W))$
52	14	cdleme31fv2	$⊢ (X \in B \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) = X$
53	19 48 52	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to F (X) = X$
54		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
55		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to (P \neq Q \land \neg Y \underset{˙}{\leq} W)$
56		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y$
57	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (Y \in B \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y)) \to F (Y) = N \underset{˙}{\lor} (Y \underset{˙}{\land} W)$
58	54 27 55 33 56 57	syl122anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to F (Y) = N \underset{˙}{\lor} (Y \underset{˙}{\land} W)$
59	51 53 58	3brtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to F (X) \underset{˙}{\leq} F (Y)$

Metamath Proof Explorer

Theorem cdleme32e

Proof