cdleme27a

Description: Part of proof of Lemma E in Crawley p. 113. cdleme26f with s and t swapped (this case is not mentioned by them). If s <_ t \/ v, then f(s) <_ f_s(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 3-Feb-2013)

	Ref	Expression
Hypotheses	cdleme26.b	$⊢ B = {Base}_{K}$
	cdleme26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme26.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme26.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme26.a	$⊢ A = Atoms (K)$
	cdleme26.h	$⊢ H = LHyp (K)$
	cdleme27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme27.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme27.z	$⊢ Z = (z \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((s \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.d	$⊢ D = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N))$
	cdleme27.c	$⊢ C = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F)$
	cdleme27.g	$⊢ G = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme27.o	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((t \underset{˙}{\lor} z) \underset{˙}{\land} W))$
	cdleme27.e	$⊢ E = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = O))$
	cdleme27.y	$⊢ Y = if (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q), E, G)$
Assertion	cdleme27a	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V)$

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	$⊢ B = {Base}_{K}$
2		cdleme26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme26.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme26.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme26.a	$⊢ A = Atoms (K)$
6		cdleme26.h	$⊢ H = LHyp (K)$
7		cdleme27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme27.f	$⊢ F = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme27.z	$⊢ Z = (z \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} z) \underset{˙}{\land} W))$
10		cdleme27.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((s \underset{˙}{\lor} z) \underset{˙}{\land} W))$
11		cdleme27.d	$⊢ D = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = N))$
12		cdleme27.c	$⊢ C = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F)$
13		cdleme27.g	$⊢ G = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
14		cdleme27.o	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Z \underset{˙}{\lor} ((t \underset{˙}{\lor} z) \underset{˙}{\land} W))$
15		cdleme27.e	$⊢ E = (ι u \in B \| \forall z \in A ((\neg z \underset{˙}{\leq} W \land \neg z \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = O))$
16		cdleme27.y	$⊢ Y = if (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q), E, G)$
17		simp211	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to (K \in HL \land W \in H)$
18		simp221	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
19		simp222	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
20		simp213	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
21		simp223	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
22		simp23r	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to (V \in A \land V \underset{˙}{\leq} W)$
23		simp212	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to P \neq Q$
24		simp1l	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
25		simp1r	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
26	23 24 25	3jca	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to (P \neq Q \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
27		simp3	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q$
28	1 2 3 4 5 6 7 9 10 14 11 15	cdleme26ee	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W) \land (V \in A \land V \underset{˙}{\leq} W)) \land ((P \neq Q \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q)) \to D \underset{˙}{\leq} (E \underset{˙}{\lor} V)$
29	17 18 19 20 21 22 26 27 28	syl332anc	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q) \to D \underset{˙}{\leq} (E \underset{˙}{\lor} V)$
30	29	3expia	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (t \underset{˙}{\lor} V = P \underset{˙}{\lor} Q \to D \underset{˙}{\leq} (E \underset{˙}{\lor} V))$
31		simp1r	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
32		simp11l	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to K \in HL$
33	32	3ad2ant2	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to K \in HL$
34		simp13l	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to s \in A$
35	34	3ad2ant2	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to s \in A$
36		simp23l	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to t \in A$
37	36	3ad2ant2	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to t \in A$
38		simp3ll	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to s \neq t$
39	38	3ad2ant2	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to s \neq t$
40	35 37 39	3jca	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to (s \in A \land t \in A \land s \neq t)$
41		simp21l	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to P \in A$
42	41	3ad2ant2	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to P \in A$
43		simp22l	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to Q \in A$
44	43	3ad2ant2	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to Q \in A$
45		simp212	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to P \neq Q$
46		simp3rl	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to V \in A$
47	46	3ad2ant2	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to V \in A$
48		simp3	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q$
49		simp3lr	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to s \underset{˙}{\leq} (t \underset{˙}{\lor} V)$
50	49	3ad2ant2	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to s \underset{˙}{\leq} (t \underset{˙}{\lor} V)$
51		simp1l	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
52	48 50 51	3jca	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to (t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
53	2 3 4 5 6	cdleme22b	$⊢ ((K \in HL \land (s \in A \land t \in A \land s \neq t)) \land (P \in A \land Q \in A \land P \neq Q) \land (V \in A \land (t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
54	33 40 42 44 45 47 52 53	syl232anc	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
55	31 54	pm2.21dd	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \land t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q) \to D \underset{˙}{\leq} (E \underset{˙}{\lor} V)$
56	55	3expia	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (t \underset{˙}{\lor} V \neq P \underset{˙}{\lor} Q \to D \underset{˙}{\leq} (E \underset{˙}{\lor} V))$
57	30 56	pm2.61dne	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to D \underset{˙}{\leq} (E \underset{˙}{\lor} V)$
58		iftrue	$⊢ s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F) = D$
59	12 58	syl5eq	$⊢ s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to C = D$
60	59	ad2antrr	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to C = D$
61		iftrue	$⊢ t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to if (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q), E, G) = E$
62	16 61	syl5eq	$⊢ t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to Y = E$
63	62	oveq1d	$⊢ t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to Y \underset{˙}{\lor} V = E \underset{˙}{\lor} V$
64	63	ad2antlr	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to Y \underset{˙}{\lor} V = E \underset{˙}{\lor} V$
65	57 60 64	3brtr4d	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V)$
66	65	ex	$⊢ (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V))$
67		simpr11	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (K \in HL \land W \in H)$
68		simpr12	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to P \neq Q$
69		simpll	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
70	68 69	jca	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (P \neq Q \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
71		simpr23	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
72		simpr21	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
73		simpr22	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
74		simpr13	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
75		simplr	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
76		simpr3l	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V))$
77		simpr3r	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (V \in A \land V \underset{˙}{\leq} W)$
78		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W)) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
79		eqid	$⊢ (ι u \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W)))) = (ι u \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))))$
80	9 10 13 78 11 79	cdleme25cv	$⊢ D = (ι u \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))))$
81	1 2 3 4 5 6 7 13 78 80	cdleme26f	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land (\neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land (s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to D \underset{˙}{\leq} (G \underset{˙}{\lor} V)$
82	67 70 71 72 73 74 75 76 77 81	syl333anc	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to D \underset{˙}{\leq} (G \underset{˙}{\lor} V)$
83	59	ad2antrr	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to C = D$
84		iffalse	$⊢ \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to if (t \underset{˙}{\leq} (P \underset{˙}{\lor} Q), E, G) = G$
85	16 84	syl5eq	$⊢ \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to Y = G$
86	85	oveq1d	$⊢ \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to Y \underset{˙}{\lor} V = G \underset{˙}{\lor} V$
87	86	ad2antlr	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to Y \underset{˙}{\lor} V = G \underset{˙}{\lor} V$
88	82 83 87	3brtr4d	$⊢ ((s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V)$
89	88	ex	$⊢ (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V))$
90		simpr11	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (K \in HL \land W \in H)$
91		simpr12	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to P \neq Q$
92		simplr	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
93	91 92	jca	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (P \neq Q \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
94		simpr13	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
95		simpr21	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
96		simpr22	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
97		simpr23	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
98		simpll	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
99		simpr3l	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V))$
100		simpr3r	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (V \in A \land V \underset{˙}{\leq} W)$
101		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((t \underset{˙}{\lor} s) \underset{˙}{\land} W)) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((t \underset{˙}{\lor} s) \underset{˙}{\land} W))$
102		eqid	$⊢ (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((t \underset{˙}{\lor} s) \underset{˙}{\land} W)))) = (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((t \underset{˙}{\lor} s) \underset{˙}{\land} W))))$
103	9 14 8 101 15 102	cdleme25cv	$⊢ E = (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((t \underset{˙}{\lor} s) \underset{˙}{\land} W))))$
104	1 2 3 4 5 6 7 8 101 103	cdleme26f2	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land (s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to F \underset{˙}{\leq} (E \underset{˙}{\lor} V)$
105	90 93 94 95 96 97 98 99 100 104	syl333anc	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to F \underset{˙}{\leq} (E \underset{˙}{\lor} V)$
106		iffalse	$⊢ \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), D, F) = F$
107	12 106	syl5eq	$⊢ \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to C = F$
108	107	ad2antrr	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to C = F$
109	63	ad2antlr	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to Y \underset{˙}{\lor} V = E \underset{˙}{\lor} V$
110	105 108 109	3brtr4d	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V)$
111	110	ex	$⊢ (\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V))$
112		simpr11	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (K \in HL \land W \in H)$
113		simpr23	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
114		simplr	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
115		simpll	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
116		simpr12	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to P \neq Q$
117	114 115 116	3jca	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (\neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)$
118		simpr21	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
119		simpr22	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
120		simpr13	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
121		simpr3l	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V))$
122		simpr3r	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to (V \in A \land V \underset{˙}{\leq} W)$
123	2 3 4 5 6 7 8 13	cdleme22g	$⊢ (((K \in HL \land W \in H) \land (t \in A \land \neg t \underset{˙}{\leq} W) \land (\neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land (s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to F \underset{˙}{\leq} (G \underset{˙}{\lor} V)$
124	112 113 117 118 119 120 121 122 123	syl323anc	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to F \underset{˙}{\leq} (G \underset{˙}{\lor} V)$
125	107	ad2antrr	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to C = F$
126	86	ad2antlr	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to Y \underset{˙}{\lor} V = G \underset{˙}{\lor} V$
127	124 125 126	3brtr4d	$⊢ ((\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W)))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V)$
128	127	ex	$⊢ (\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V))$
129	66 89 111 128	4cases	$⊢ (((K \in HL \land W \in H) \land P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land ((s \neq t \land s \underset{˙}{\leq} (t \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to C \underset{˙}{\leq} (Y \underset{˙}{\lor} V)$

Metamath Proof Explorer

Theorem cdleme27a

Proof