cdlemefs32fvaN

Description: Part of proof of Lemma E in Crawley p. 113. Value of F at an atom not under W . TODO: FIX COMMENT. TODO: consolidate uses of lhpmat here and elsewhere, and presence/absence of s .<_ ( P .\/ Q ) term. Also, why can proof be shortened with cdleme27cl ? What is difference from cdlemefs27cl ? (Contributed by NM, 29-Mar-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemefs32.b	$⊢ B = {Base}_{K}$
	cdlemefs32.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemefs32.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemefs32.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemefs32.a	$⊢ A = Atoms (K)$
	cdlemefs32.h	$⊢ H = LHyp (K)$
	cdlemefs32.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdlemefs32.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemefs32.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemefs32.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
	cdlemefs32.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
	cdleme29fs.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
Assertion	cdlemefs32fvaN	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / x ⦌ O = ⦋ R / s ⦌ N$

Proof

Step	Hyp	Ref	Expression
1		cdlemefs32.b	$⊢ B = {Base}_{K}$
2		cdlemefs32.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemefs32.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemefs32.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemefs32.a	$⊢ A = Atoms (K)$
6		cdlemefs32.h	$⊢ H = LHyp (K)$
7		cdlemefs32.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemefs32.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemefs32.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemefs32.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
11		cdlemefs32.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
12		cdleme29fs.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
13		breq1	$⊢ s = R \to (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
14		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
15		simp3l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to s \in A$
16		simp3rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to \neg s \underset{˙}{\leq} W$
17	15 16	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
18		simp3rr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
19		simp2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to P \neq Q$
20	1 2 3 4 5 6 7 8 9 10 11	cdlemefs27cl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \to N \in B$
21	14 17 18 19 20	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to N \in B$
22	1 2 3 4 5 6 7 8 9 10 11	cdlemefs32snb	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / s ⦌ N \in B$
23	1 2 3 4 5 6 13 21 22 12	cdlemefrs32fva	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / x ⦌ O = ⦋ R / s ⦌ N$

Metamath Proof Explorer

Theorem cdlemefs32fvaN

Proof