cdlemk50

Description: Part of proof of Lemma K of Crawley p. 118. Line 6, p. 120. G , I stand for g, h. X represents tau. TODO: Combine into cdlemk52 ? (Contributed by NM, 23-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk5.b	$⊢ B = {Base}_{K}$
	cdlemk5.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk5.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk5.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk5.a	$⊢ A = Atoms (K)$
	cdlemk5.h	$⊢ H = LHyp (K)$
	cdlemk5.t	$⊢ T = LTrn (K) (W)$
	cdlemk5.r	$⊢ R = trL (K) (W)$
	cdlemk5.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
	cdlemk5.y	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
	cdlemk5.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	cdlemk50	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk5.b	$⊢ B = {Base}_{K}$
2		cdlemk5.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk5.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk5.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk5.a	$⊢ A = Atoms (K)$
6		cdlemk5.h	$⊢ H = LHyp (K)$
7		cdlemk5.t	$⊢ T = LTrn (K) (W)$
8		cdlemk5.r	$⊢ R = trL (K) (W)$
9		cdlemk5.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
10		cdlemk5.y	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
11		cdlemk5.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	1 2 3 4 5 6 7 8 9 10 11	cdlemk49	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X))$
13	1 2 3 4 5 6 7 8 9 10 11	cdlemk48	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))$
14		simp11l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to K \in HL$
15	14	hllatd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to K \in Lat$
16		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (K \in HL \land W \in H)$
17		simp12	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (F \in T \land F \neq {I ↾}_{B})$
18		simp13	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (G \in T \land G \neq {I ↾}_{B})$
19		simp21	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to N \in T$
20		simp22	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
21		simp23	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (F) = R (N)$
22	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ G / g ⦌ X \in T$
23	16 17 18 19 20 21 22	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X \in T$
24		simp3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (I \in T \land I \neq {I ↾}_{B})$
25	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (I \in T \land I \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ I / g ⦌ X \in T$
26	16 17 24 19 20 21 25	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X \in T$
27	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T \land ⦋ I / g ⦌ X \in T) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X \in T$
28	16 23 26 27	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X \in T$
29		simp22l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to P \in A$
30	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X \in T \land P \in A) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in A$
31	16 28 29 30	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in A$
32	1 5	atbase	$⊢ (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in A \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in B$
33	31 32	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in B$
34	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T \land P \in A) \to ⦋ G / g ⦌ X (P) \in A$
35	16 23 29 34	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \in A$
36	1 5	atbase	$⊢ ⦋ G / g ⦌ X (P) \in A \to ⦋ G / g ⦌ X (P) \in B$
37	35 36	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \in B$
38	1 6 7 8	trlcl	$⊢ ((K \in HL \land W \in H) \land ⦋ I / g ⦌ X \in T) \to R (⦋ I / g ⦌ X) \in B$
39	16 26 38	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (⦋ I / g ⦌ X) \in B$
40	1 3	latjcl	$⊢ (K \in Lat \land ⦋ G / g ⦌ X (P) \in B \land R (⦋ I / g ⦌ X) \in B) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X) \in B$
41	15 37 39 40	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X) \in B$
42	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land ⦋ I / g ⦌ X \in T \land P \in A) \to ⦋ I / g ⦌ X (P) \in A$
43	16 26 29 42	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X (P) \in A$
44	1 5	atbase	$⊢ ⦋ I / g ⦌ X (P) \in A \to ⦋ I / g ⦌ X (P) \in B$
45	43 44	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X (P) \in B$
46	1 6 7 8	trlcl	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T) \to R (⦋ G / g ⦌ X) \in B$
47	16 23 46	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (⦋ G / g ⦌ X) \in B$
48	1 3	latjcl	$⊢ (K \in Lat \land ⦋ I / g ⦌ X (P) \in B \land R (⦋ G / g ⦌ X) \in B) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X) \in B$
49	15 45 47 48	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X) \in B$
50	1 2 4	latlem12	$⊢ (K \in Lat \land ((⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in B \land ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X) \in B \land ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X) \in B)) \to (((⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \land (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))) \leftrightarrow (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))))$
51	15 33 41 49 50	syl13anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (((⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \land (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))) \leftrightarrow (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))))$
52	12 13 51	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)))$

Metamath Proof Explorer

Theorem cdlemk50

Proof