cdleme48b

	Ref	Expression
Hypotheses	cdlemef46.b	$⊢ B = {Base}_{K}$
	cdlemef46.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemef46.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemef46.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemef46.a	$⊢ A = Atoms (K)$
	cdlemef46.h	$⊢ H = LHyp (K)$
	cdlemef46.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdlemef46.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemefs46.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemef46.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
Assertion	cdleme48b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) \underset{˙}{\land} W = X \underset{˙}{\land} W$

Proof

Step	Hyp	Ref	Expression
1		cdlemef46.b	$⊢ B = {Base}_{K}$
2		cdlemef46.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemef46.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemef46.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemef46.a	$⊢ A = Atoms (K)$
6		cdlemef46.h	$⊢ H = LHyp (K)$
7		cdlemef46.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemef46.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemefs46.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemef46.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
11	1 2 3 4 5 6 7 8 9 10	cdleme48fv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) = F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$
12	11	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) \underset{˙}{\land} W = (F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\land} W$
13		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (K \in HL \land W \in H)$
14		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
15		simp3l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
16	1 2 3 4 5 6 7 8 9 10	cdleme46fvaw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \to (F (S) \in A \land \neg F (S) \underset{˙}{\leq} W)$
17	14 15 16	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (F (S) \in A \land \neg F (S) \underset{˙}{\leq} W)$
18		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to X \in B$
19	1 2 3 4 5 6	lhpelim	$⊢ ((K \in HL \land W \in H) \land (F (S) \in A \land \neg F (S) \underset{˙}{\leq} W) \land X \in B) \to (F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\land} W = X \underset{˙}{\land} W$
20	13 17 18 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)) \underset{˙}{\land} W = X \underset{˙}{\land} W$
21	12 20	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) \underset{˙}{\land} W = X \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdleme48b

Proof