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Metamath Proof Explorer

Theorem cdlemefr31fv1

Description: Value of ( FR ) when -. R .<_ ( P .\/ Q ) . TODO This may be useful for shortening others that now use riotasv 3d . TODO: FIX COMMENT. (Contributed by NM, 30-Mar-2013)

		Ref	Expression
	Hypotheses	cdlemefr27.b	$⊢ B = {Base}_{K}$
		cdlemefr27.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemefr27.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemefr27.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemefr27.a	$⊢ A = Atoms (K)$
		cdlemefr27.h	$⊢ H = LHyp (K)$
		cdlemefr27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
		cdlemefr27.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
		cdlemefr27.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
		cdleme29fr.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
		cdleme29fr.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
		cdleme43frv.x	$⊢ X = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
	Assertion	cdlemefr31fv1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F (R) = X$

Proof

Step	Hyp	Ref	Expression
1		cdlemefr27.b	$⊢ B = {Base}_{K}$
2		cdlemefr27.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemefr27.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemefr27.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemefr27.a	$⊢ A = Atoms (K)$
6		cdlemefr27.h	$⊢ H = LHyp (K)$
7		cdlemefr27.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemefr27.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdlemefr27.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
10		cdleme29fr.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
11		cdleme29fr.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
12		cdleme43frv.x	$⊢ X = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
13	1 2 3 4 5 6 7 8 9 10 11	cdlemefr32fva1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F (R) = ⦋ R / s ⦌ N$
14		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \in A$
15		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
16	8 9 12	cdleme31sn2	$⊢ (R \in A \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / s ⦌ N = X$
17	14 15 16	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ⦋ R / s ⦌ N = X$
18	13 17	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F (R) = X$