cdlemk51

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	cdlemk51	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)))$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (K \in HL \land W \in H)$
13		simp12	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (F \in T \land F \neq {I ↾}_{B})$
14		simp3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (I \in T \land I \neq {I ↾}_{B})$
15		simp21	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to N \in T$
16		simp22	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
17		simp23	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (F) = R (N)$
18	1 2 3 4 5 6 7 8 9 10 11	cdlemk39s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (I \in T \land I \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (⦋ I / g ⦌ X) \underset{˙}{\leq} R (I)$
19	12 13 14 15 16 17 18	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (⦋ I / g ⦌ X) \underset{˙}{\leq} R (I)$
20		simp11l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to K \in HL$
21	20	hllatd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to K \in Lat$
22	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (I \in T \land I \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ I / g ⦌ X \in T$
23	12 13 14 15 16 17 22	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X \in T$
24	1 6 7 8	trlcl	$⊢ ((K \in HL \land W \in H) \land ⦋ I / g ⦌ X \in T) \to R (⦋ I / g ⦌ X) \in B$
25	12 23 24	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (⦋ I / g ⦌ X) \in B$
26		simp3l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to I \in T$
27		simp3r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to I \neq {I ↾}_{B}$
28	1 5 6 7 8	trlnidat	$⊢ ((K \in HL \land W \in H) \land I \in T \land I \neq {I ↾}_{B}) \to R (I) \in A$
29	12 26 27 28	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (I) \in A$
30	1 5	atbase	$⊢ R (I) \in A \to R (I) \in B$
31	29 30	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (I) \in B$
32		simp13	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (G \in T \land G \neq {I ↾}_{B})$
33	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ G / g ⦌ X \in T$
34	12 13 32 15 16 17 33	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X \in T$
35		simp22l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to P \in A$
36	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T \land P \in A) \to ⦋ G / g ⦌ X (P) \in A$
37	12 34 35 36	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \in A$
38	1 5	atbase	$⊢ ⦋ G / g ⦌ X (P) \in A \to ⦋ G / g ⦌ X (P) \in B$
39	37 38	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \in B$
40	1 2 3	latjlej2	$⊢ (K \in Lat \land (R (⦋ I / g ⦌ X) \in B \land R (I) \in B \land ⦋ G / g ⦌ X (P) \in B)) \to (R (⦋ I / g ⦌ X) \underset{˙}{\leq} R (I) \to (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)))$
41	21 25 31 39 40	syl13anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (R (⦋ I / g ⦌ X) \underset{˙}{\leq} R (I) \to (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)))$
42	19 41	mpd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I))$
43	1 2 3 4 5 6 7 8 9 10 11	cdlemk39s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G)$
44	12 13 32 15 16 17 43	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G)$
45	1 6 7 8	trlcl	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T) \to R (⦋ G / g ⦌ X) \in B$
46	12 34 45	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (⦋ G / g ⦌ X) \in B$
47		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to G \in T$
48		simp13r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to G \neq {I ↾}_{B}$
49	1 5 6 7 8	trlnidat	$⊢ ((K \in HL \land W \in H) \land G \in T \land G \neq {I ↾}_{B}) \to R (G) \in A$
50	12 47 48 49	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (G) \in A$
51	1 5	atbase	$⊢ R (G) \in A \to R (G) \in B$
52	50 51	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to R (G) \in B$
53	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land ⦋ I / g ⦌ X \in T \land P \in A) \to ⦋ I / g ⦌ X (P) \in A$
54	12 23 35 53	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X (P) \in A$
55	1 5	atbase	$⊢ ⦋ I / g ⦌ X (P) \in A \to ⦋ I / g ⦌ X (P) \in B$
56	54 55	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X (P) \in B$
57	1 2 3	latjlej2	$⊢ (K \in Lat \land (R (⦋ G / g ⦌ X) \in B \land R (G) \in B \land ⦋ I / g ⦌ X (P) \in B)) \to (R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G) \to (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)))$
58	21 46 52 56 57	syl13anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (R (⦋ G / g ⦌ X) \underset{˙}{\leq} R (G) \to (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)))$
59	44 58	mpd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$
60	1 3	latjcl	$⊢ (K \in Lat \land ⦋ G / g ⦌ X (P) \in B \land R (⦋ I / g ⦌ X) \in B) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X) \in B$
61	21 39 25 60	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X) \in B$
62	1 3 5	hlatjcl	$⊢ (K \in HL \land ⦋ G / g ⦌ X (P) \in A \land R (I) \in A) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \in B$
63	20 37 29 62	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \in B$
64	1 3	latjcl	$⊢ (K \in Lat \land ⦋ I / g ⦌ X (P) \in B \land R (⦋ G / g ⦌ X) \in B) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X) \in B$
65	21 56 46 64	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X) \in B$
66	1 3 5	hlatjcl	$⊢ (K \in HL \land ⦋ I / g ⦌ X (P) \in A \land R (G) \in A) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G) \in B$
67	20 54 50 66	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G) \in B$
68	1 2 4	latmlem12	$⊢ (K \in Lat \land (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X) \in B \land ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \in B) \land (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X) \in B \land ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G) \in B)) \to (((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \land (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))) \to ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))))$
69	21 61 63 65 67 68	syl122anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \land (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X)) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))) \to ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))))$
70	42 59 69	mp2and	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (⦋ G / g ⦌ X))) \underset{˙}{\leq} ((⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)))$

Metamath Proof Explorer

Theorem cdlemk51

Proof