cdleme28c

Description: Part of proof of Lemma E in Crawley p. 113. Eliminate the s =/= t antecedent in cdleme28b . TODO: FIX COMMENT. (Contributed by NM, 6-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	cdleme28c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$

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	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16	cdleme27b	$⊢ s = t \to C = Y$
18	17	oveq1d	$⊢ s = t \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$
19	18	adantl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s = t) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$
20		simpl11	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to (K \in HL \land W \in H)$
21		simpl12	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
22		simpl13	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
23		simpl21	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to P \neq Q$
24		simpl22	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
25		simpl23	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to (t \in A \land \neg t \underset{˙}{\leq} W)$
26		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to s \neq t$
27		simpl31	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
28		simpl32	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
29	27 28	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
30		simpl33	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
31	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16	cdleme28b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \neq t \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X) \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$
32	20 21 22 23 24 25 26 29 30 31	syl333anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \land s \neq t) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$
33	19 32	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (s \in A \land \neg s \underset{˙}{\leq} W) \land (t \in A \land \neg t \underset{˙}{\leq} W)) \land (s \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land t \underset{˙}{\lor} (X \underset{˙}{\land} W) = X \land (X \in B \land \neg X \underset{˙}{\leq} W))) \to C \underset{˙}{\lor} (X \underset{˙}{\land} W) = Y \underset{˙}{\lor} (X \underset{˙}{\land} W)$

Metamath Proof Explorer

Theorem cdleme28c

Proof