cdlemefrs32fva1

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013)

	Ref	Expression
Hypotheses	cdlemefrs27.b	$⊢ B = {Base}_{K}$
	cdlemefrs27.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemefrs27.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemefrs27.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemefrs27.a	$⊢ A = Atoms (K)$
	cdlemefrs27.h	$⊢ H = LHyp (K)$
	cdlemefrs27.eq	$⊢ s = R \to (φ \leftrightarrow ψ)$
	cdlemefrs27.nb	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land φ))) \to N \in B$
	cdlemefrs27.rnb	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to ⦋ R / s ⦌ N \in B$
	cdleme29frs.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
	cdleme29frs.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
Assertion	cdlemefrs32fva1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to F (R) = ⦋ R / s ⦌ N$

Proof

Step	Hyp	Ref	Expression
1		cdlemefrs27.b	$⊢ B = {Base}_{K}$
2		cdlemefrs27.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemefrs27.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemefrs27.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemefrs27.a	$⊢ A = Atoms (K)$
6		cdlemefrs27.h	$⊢ H = LHyp (K)$
7		cdlemefrs27.eq	$⊢ s = R \to (φ \leftrightarrow ψ)$
8		cdlemefrs27.nb	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (s \in A \land (\neg s \underset{˙}{\leq} W \land φ))) \to N \in B$
9		cdlemefrs27.rnb	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to ⦋ R / s ⦌ N \in B$
10		cdleme29frs.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		cdleme29frs.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
12		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to R \in A$
13	1 5	atbase	$⊢ R \in A \to R \in B$
14	12 13	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to R \in B$
15		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to P \neq Q$
16		simp2rr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to \neg R \underset{˙}{\leq} W$
17	10 11	cdleme31fv1s	$⊢ (R \in B \land (P \neq Q \land \neg R \underset{˙}{\leq} W)) \to F (R) = ⦋ R / x ⦌ O$
18	14 15 16 17	syl12anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to F (R) = ⦋ R / x ⦌ O$
19	1 2 3 4 5 6 7 8 9 10	cdlemefrs32fva	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to ⦋ R / x ⦌ O = ⦋ R / s ⦌ N$
20	18 19	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ψ) \to F (R) = ⦋ R / s ⦌ N$

Metamath Proof Explorer

Theorem cdlemefrs32fva1

Proof