cdleme48bw

	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	cdleme48bw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to \neg F (X) \underset{˙}{\leq} W$

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		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
12		simp3l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
13	1 2 3 4 5 6 7 8 9 10	cdleme46fvaw	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \to (F (S) \in A \land \neg F (S) \underset{˙}{\leq} W)$
14	11 12 13	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (F (S) \in A \land \neg F (S) \underset{˙}{\leq} W)$
15	14	simprd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to \neg F (S) \underset{˙}{\leq} W$
16		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to K \in HL$
17	16	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to K \in Lat$
18	14	simpld	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (S) \in A$
19	1 5	atbase	$⊢ F (S) \in A \to F (S) \in B$
20	18 19	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (S) \in B$
21		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to X \in B$
22		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to W \in H$
23	1 6	lhpbase	$⊢ W \in H \to W \in B$
24	22 23	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to W \in B$
25	1 4	latmcl	$⊢ (K \in Lat \land X \in B \land W \in B) \to X \underset{˙}{\land} W \in B$
26	17 21 24 25	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to X \underset{˙}{\land} W \in B$
27	1 2 3	latlej1	$⊢ (K \in Lat \land F (S) \in B \land X \underset{˙}{\land} W \in B) \to F (S) \underset{˙}{\leq} (F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W))$
28	17 20 26 27	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (S) \underset{˙}{\leq} (F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W))$
29	1 2 3 4 5 6 7 8 9 10	cdleme48fv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) = F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$
30	28 29	breqtrrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (S) \underset{˙}{\leq} F (X)$
31		vex	$⊢ s \in V$
32		eqid	$⊢ (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W)) = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
33	8 32	cdleme31sc	$⊢ s \in V \to ⦋ s / t ⦌ D = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
34	31 33	ax-mp	$⊢ ⦋ s / t ⦌ D = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
35		eqid	$⊢ (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)) = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
36		eqid	$⊢ if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D)$
37		eqid	$⊢ (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))) = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
38	1 2 3 4 5 6 7 34 8 9 35 36 37 10	cdleme32fvcl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land X \in B) \to F (X) \in B$
39	11 21 38	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) \in B$
40	1 2	lattr	$⊢ (K \in Lat \land (F (S) \in B \land F (X) \in B \land W \in B)) \to ((F (S) \underset{˙}{\leq} F (X) \land F (X) \underset{˙}{\leq} W) \to F (S) \underset{˙}{\leq} W)$
41	17 20 39 24 40	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to ((F (S) \underset{˙}{\leq} F (X) \land F (X) \underset{˙}{\leq} W) \to F (S) \underset{˙}{\leq} W)$
42	30 41	mpand	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to (F (X) \underset{˙}{\leq} W \to F (S) \underset{˙}{\leq} W)$
43	15 42	mtod	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to \neg F (X) \underset{˙}{\leq} W$

Metamath Proof Explorer

Theorem cdleme48bw

Proof