cdlemk39

Description: Part of proof of Lemma K of Crawley p. 118. Line 31, p. 119. Trace-preserving property of tau, represented by X . (Contributed by NM, 19-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk4.b	$⊢ B = {Base}_{K}$
	cdlemk4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk4.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk4.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk4.a	$⊢ A = Atoms (K)$
	cdlemk4.h	$⊢ H = LHyp (K)$
	cdlemk4.t	$⊢ T = LTrn (K) (W)$
	cdlemk4.r	$⊢ R = trL (K) (W)$
	cdlemk4.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
	cdlemk4.y	$⊢ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
	cdlemk4.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y))$
Assertion	cdlemk39	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (X) \underset{˙}{\leq} R (G)$

Proof

Step	Hyp	Ref	Expression
1		cdlemk4.b	$⊢ B = {Base}_{K}$
2		cdlemk4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk4.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk4.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk4.a	$⊢ A = Atoms (K)$
6		cdlemk4.h	$⊢ H = LHyp (K)$
7		cdlemk4.t	$⊢ T = LTrn (K) (W)$
8		cdlemk4.r	$⊢ R = trL (K) (W)$
9		cdlemk4.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
10		cdlemk4.y	$⊢ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
11		cdlemk4.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y))$
12		simp1l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to K \in HL$
13		simp3ll	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to P \in A$
14		simp1	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (K \in HL \land W \in H)$
15		simp22l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to G \in T$
16		simp22r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to G \neq {I ↾}_{B}$
17	1 5 6 7 8	trlnidat	$⊢ ((K \in HL \land W \in H) \land G \in T \land G \neq {I ↾}_{B}) \to R (G) \in A$
18	14 15 16 17	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (G) \in A$
19	2 3 5	hlatlej1	$⊢ (K \in HL \land P \in A \land R (G) \in A) \to P \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$
20	12 13 18 19	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to P \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$
21	1 2 3 4 5 6 7 8 9 10 11	cdlemk38	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$
22	12	hllatd	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to K \in Lat$
23	1 5	atbase	$⊢ P \in A \to P \in B$
24	13 23	syl	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to P \in B$
25	1 2 3 4 5 6 7 8 9 10 11	cdlemk35	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X \in T$
26	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land X \in T \land P \in A) \to X (P) \in A$
27	14 25 13 26	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X (P) \in A$
28	1 5	atbase	$⊢ X (P) \in A \to X (P) \in B$
29	27 28	syl	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X (P) \in B$
30	1 3 5	hlatjcl	$⊢ (K \in HL \land P \in A \land R (G) \in A) \to P \underset{˙}{\lor} R (G) \in B$
31	12 13 18 30	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to P \underset{˙}{\lor} R (G) \in B$
32	1 2 3	latjle12	$⊢ (K \in Lat \land (P \in B \land X (P) \in B \land P \underset{˙}{\lor} R (G) \in B)) \to ((P \underset{˙}{\leq} (P \underset{˙}{\lor} R (G)) \land X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))) \leftrightarrow (P \underset{˙}{\lor} X (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G)))$
33	22 24 29 31 32	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ((P \underset{˙}{\leq} (P \underset{˙}{\lor} R (G)) \land X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))) \leftrightarrow (P \underset{˙}{\lor} X (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G)))$
34	20 21 33	mpbi2and	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (P \underset{˙}{\lor} X (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$
35	1 3 5	hlatjcl	$⊢ (K \in HL \land P \in A \land X (P) \in A) \to P \underset{˙}{\lor} X (P) \in B$
36	12 13 27 35	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to P \underset{˙}{\lor} X (P) \in B$
37		simp1r	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to W \in H$
38	1 6	lhpbase	$⊢ W \in H \to W \in B$
39	37 38	syl	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to W \in B$
40	1 2 4	latmlem1	$⊢ (K \in Lat \land (P \underset{˙}{\lor} X (P) \in B \land P \underset{˙}{\lor} R (G) \in B \land W \in B)) \to ((P \underset{˙}{\lor} X (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G)) \to ((P \underset{˙}{\lor} X (P)) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} W))$
41	22 36 31 39 40	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ((P \underset{˙}{\lor} X (P)) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G)) \to ((P \underset{˙}{\lor} X (P)) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} W))$
42	34 41	mpd	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ((P \underset{˙}{\lor} X (P)) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} W)$
43		simp3l	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
44	2 3 4 5 6 7 8	trlval2	$⊢ ((K \in HL \land W \in H) \land X \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (X) = (P \underset{˙}{\lor} X (P)) \underset{˙}{\land} W$
45	14 25 43 44	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (X) = (P \underset{˙}{\lor} X (P)) \underset{˙}{\land} W$
46	2 3 4 5 6 7 8	trlval5	$⊢ ((K \in HL \land W \in H) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (G) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} W$
47	14 15 43 46	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (G) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} W$
48	42 45 47	3brtr4d	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (X) \underset{˙}{\leq} R (G)$

Metamath Proof Explorer

Theorem cdlemk39

Proof