cdlemk52

Description: Part of proof of Lemma K of Crawley p. 118. Line 6, p. 120. G , I stand for g, h. X represents tau. (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	cdlemk52	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) = ⦋ (G \circ I) / g ⦌ X (P)$

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		simp11l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to K \in HL$
13	12	hllatd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to K \in Lat$
14		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (K \in HL \land W \in H)$
15		simp12	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (F \in T \land F \neq {I ↾}_{B})$
16		simp13	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (G \in T \land G \neq {I ↾}_{B})$
17		simp21	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to N \in T$
18		simp22	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
19		simp23	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (F) = R (N)$
20	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ G / g ⦌ X \in T$
21	14 15 16 17 18 19 20	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X \in T$
22		simp31	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to I \in T$
23		simp32	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to I \neq {I ↾}_{B}$
24	22 23	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (I \in T \land I \neq {I ↾}_{B})$
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	14 15 24 17 18 19 25	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X \in T$
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	14 21 26 27	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X \in T$
29		simp22l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to P \in A$
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	14 28 29 30	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in A$
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	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in B$
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	14 21 29 34	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X (P) \in A$
36	1 5	atbase	$⊢ ⦋ G / g ⦌ X (P) \in A \to ⦋ G / g ⦌ X (P) \in B$
37	35 36	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X (P) \in B$
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	14 26 38	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (⦋ I / g ⦌ X) \in B$
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	13 37 39 40	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X) \in B$
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	14 26 29 42	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X (P) \in A$
44	1 5	atbase	$⊢ ⦋ I / g ⦌ X (P) \in A \to ⦋ I / g ⦌ X (P) \in B$
45	43 44	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X (P) \in B$
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	14 21 46	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (⦋ G / g ⦌ X) \in B$
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	13 45 47 48	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X) \in B$
50	1 4	latmcl	$⊢ (K \in Lat \land ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X) \in B \land ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X) \in B) \to (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)) \in B$
51	13 41 49 50	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)) \in B$
52		simp11r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to W \in H$
53	1 5 6 7 8	trlnidat	$⊢ ((K \in HL \land W \in H) \land I \in T \land I \neq {I ↾}_{B}) \to R (I) \in A$
54	12 52 22 23 53	syl211anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (I) \in A$
55	1 3 5	hlatjcl	$⊢ (K \in HL \land ⦋ G / g ⦌ X (P) \in A \land R (I) \in A) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \in B$
56	12 35 54 55	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \in B$
57		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to G \in T$
58		simp13r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to G \neq {I ↾}_{B}$
59	1 5 6 7 8	trlnidat	$⊢ ((K \in HL \land W \in H) \land G \in T \land G \neq {I ↾}_{B}) \to R (G) \in A$
60	12 52 57 58 59	syl211anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (G) \in A$
61	1 3 5	hlatjcl	$⊢ (K \in HL \land ⦋ I / g ⦌ X (P) \in A \land R (G) \in A) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G) \in B$
62	12 43 60 61	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G) \in B$
63	1 4	latmcl	$⊢ (K \in Lat \land ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \in B \land ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G) \in B) \to (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)) \in B$
64	13 56 62 63	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)) \in B$
65	1 2 3 4 5 6 7 8 9 10 11	cdlemk50	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)))$
66	24 65	syld3an3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)))$
67	1 2 3 4 5 6 7 8 9 10 11	cdlemk51	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)))$
68	24 67	syld3an3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)))$
69	1 2 13 33 51 64 66 68	lattrd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)))$
70	1 2 3 4 5 6 7 8 9 10 11	cdlemk47	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X (P) = (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$
71	69 70	breqtrrd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ⦋ (G \circ I) / g ⦌ X (P)$
72		hlatl	$⊢ K \in HL \to K \in AtLat$
73	12 72	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to K \in AtLat$
74	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land G \in T \land I \in T) \to G \circ I \in T$
75	14 57 22 74	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to G \circ I \in T$
76	57 22	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (G \in T \land I \in T)$
77		simp33	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (G) \neq R (I)$
78	1 6 7 8	trlconid	$⊢ ((K \in HL \land W \in H) \land (G \in T \land I \in T) \land R (G) \neq R (I)) \to G \circ I \neq {I ↾}_{B}$
79	14 76 77 78	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to G \circ I \neq {I ↾}_{B}$
80	75 79	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (G \circ I \in T \land G \circ I \neq {I ↾}_{B})$
81	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \circ I \in T \land G \circ I \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ (G \circ I) / g ⦌ X \in T$
82	14 15 80 17 18 19 81	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X \in T$
83	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land ⦋ (G \circ I) / g ⦌ X \in T \land P \in A) \to ⦋ (G \circ I) / g ⦌ X (P) \in A$
84	14 82 29 83	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X (P) \in A$
85	2 5	atcmp	$⊢ (K \in AtLat \land (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \in A \land ⦋ (G \circ I) / g ⦌ X (P) \in A) \to ((⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ⦋ (G \circ I) / g ⦌ X (P) \leftrightarrow (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) = ⦋ (G \circ I) / g ⦌ X (P))$
86	73 31 84 85	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ((⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} ⦋ (G \circ I) / g ⦌ X (P) \leftrightarrow (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) = ⦋ (G \circ I) / g ⦌ X (P))$
87	71 86	mpbid	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) = ⦋ (G \circ I) / g ⦌ X (P)$

Metamath Proof Explorer

Theorem cdlemk52

Proof