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Metamath Proof Explorer

Theorem cdleme43fsv1sn

Description: Value of [_ R / s ]_ N when R .<_ ( P .\/ Q ) . (Contributed by NM, 30-Mar-2013)

		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	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
		cdleme43fs.y	$⊢ Y = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
		cdleme43fs.z	$⊢ Z = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Y \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	Assertion	cdleme43fsv1sn	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ⦋ R / s ⦌ N = Z$

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		cdleme43fs.y	$⊢ Y = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
13		cdleme43fs.z	$⊢ Z = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Y \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
14		eqid	$⊢ (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W)) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W))$
15		eqid	$⊢ (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W)))) = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((R \underset{˙}{\lor} t) \underset{˙}{\land} W))))$
16	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	cdleme43fsv1snlem	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ⦋ R / s ⦌ N = Z$