cdleme42h

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 8-Mar-2013)

	Ref	Expression
Hypotheses	cdleme41.b	$⊢ B = {Base}_{K}$
	cdleme41.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme41.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme41.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme41.a	$⊢ A = Atoms (K)$
	cdleme41.h	$⊢ H = LHyp (K)$
	cdleme41.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme41.d	$⊢ D = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme41.e	$⊢ E = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme41.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (E \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme41.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = G))$
	cdleme41.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, D)$
	cdleme41.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
	cdleme41.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
	cdleme34e.v	$⊢ V = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
Assertion	cdleme42h	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (S) \underset{˙}{\leq} (F (R) \underset{˙}{\lor} V)$

Proof

Step	Hyp	Ref	Expression
1		cdleme41.b	$⊢ B = {Base}_{K}$
2		cdleme41.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme41.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme41.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme41.a	$⊢ A = Atoms (K)$
6		cdleme41.h	$⊢ H = LHyp (K)$
7		cdleme41.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme41.d	$⊢ D = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme41.e	$⊢ E = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdleme41.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (E \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
11		cdleme41.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = G))$
12		cdleme41.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, D)$
13		cdleme41.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
14		cdleme41.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
15		cdleme34e.v	$⊢ V = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
16		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to K \in HL$
17	16	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to K \in Lat$
18		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
19		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to S \in A$
20	1 5	atbase	$⊢ S \in A \to S \in B$
21	19 20	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to S \in B$
22	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32fvcl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land S \in B) \to F (S) \in B$
23	18 21 22	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (S) \in B$
24		simp2ll	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to R \in A$
25	1 3 5	hlatjcl	$⊢ (K \in HL \land R \in A \land S \in A) \to R \underset{˙}{\lor} S \in B$
26	16 24 19 25	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to R \underset{˙}{\lor} S \in B$
27		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to W \in H$
28	1 6	lhpbase	$⊢ W \in H \to W \in B$
29	27 28	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to W \in B$
30	1 4	latmcl	$⊢ (K \in Lat \land R \underset{˙}{\lor} S \in B \land W \in B) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in B$
31	17 26 29 30	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in B$
32	15 31	eqeltrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to V \in B$
33	1 2 3	latlej1	$⊢ (K \in Lat \land F (S) \in B \land V \in B) \to F (S) \underset{˙}{\leq} (F (S) \underset{˙}{\lor} V)$
34	17 23 32 33	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (S) \underset{˙}{\leq} (F (S) \underset{˙}{\lor} V)$
35	3 5	hlatjcom	$⊢ (K \in HL \land R \in A \land S \in A) \to R \underset{˙}{\lor} S = S \underset{˙}{\lor} R$
36	16 24 19 35	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to R \underset{˙}{\lor} S = S \underset{˙}{\lor} R$
37	36	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W = (S \underset{˙}{\lor} R) \underset{˙}{\land} W$
38	15 37	eqtrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to V = (S \underset{˙}{\lor} R) \underset{˙}{\land} W$
39	38	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (S) \underset{˙}{\lor} V = F (S) \underset{˙}{\lor} ((S \underset{˙}{\lor} R) \underset{˙}{\land} W)$
40		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
41		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
42		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to P \neq Q$
43		eqid	$⊢ (S \underset{˙}{\lor} R) \underset{˙}{\land} W = (S \underset{˙}{\lor} R) \underset{˙}{\land} W$
44	1 2 3 4 5 6 7 8 9 10 11 12 13 14 43	cdleme42g	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land P \neq Q) \to F (S \underset{˙}{\lor} R) = F (S) \underset{˙}{\lor} ((S \underset{˙}{\lor} R) \underset{˙}{\land} W)$
45	18 40 41 42 44	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (S \underset{˙}{\lor} R) = F (S) \underset{˙}{\lor} ((S \underset{˙}{\lor} R) \underset{˙}{\land} W)$
46	39 45	eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (S) \underset{˙}{\lor} V = F (S \underset{˙}{\lor} R)$
47	36	fveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (R \underset{˙}{\lor} S) = F (S \underset{˙}{\lor} R)$
48	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	cdleme42g	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (R \underset{˙}{\lor} S) = F (R) \underset{˙}{\lor} V$
49	46 47 48	3eqtr2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (S) \underset{˙}{\lor} V = F (R) \underset{˙}{\lor} V$
50	34 49	breqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land P \neq Q) \to F (S) \underset{˙}{\leq} (F (R) \underset{˙}{\lor} V)$

Metamath Proof Explorer

Theorem cdleme42h

Proof