cdleme28

Description: Quantified version of cdleme28c . (Compare cdleme24 .) (Contributed by NM, 7-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	cdleme28	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to \forall s \in A \forall t \in A (((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$

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		simp11	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
18		simp12	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to P \neq Q$
19		simp2l	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to s \in A$
20		simp3ll	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to \neg s \underset{˙}{\leq} W$
21	19 20	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
22		simp2r	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to t \in A$
23		simp3rl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to \neg t \underset{˙}{\leq} W$
24	22 23	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
25		simp3lr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
26		simp3rr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
27		simp13	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
28	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16	cdleme28c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$
29	17 18 21 24 25 26 27 28	syl133anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land (s \in A \land t \in A) \land ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X))) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$
30	29	3exp	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to ((s \in A \land t \in A) \to (((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)))$
31	30	ralrimivv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \to \forall s \in A \forall t \in A (((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (\neg t \underset{˙}{\leq} W \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W))$

Metamath Proof Explorer

Theorem cdleme28

Proof