cdlemk47

Description: Part of proof of Lemma K of Crawley p. 118. Line 2, p. 120. G , I stand for g, h. X represents tau. (Contributed by NM, 22-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	cdlemk47	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X (P) = (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (K \in HL \land W \in H)$
14		simp12	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (F \in T \land F \neq {I ↾}_{B})$
15		simp13	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (G \in T \land G \neq {I ↾}_{B})$
16		simp21	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to N \in T$
17		simp22	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18		simp23	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (F) = R (N)$
19	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ G / g ⦌ X \in T$
20	13 14 15 16 17 18 19	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X \in T$
21	2 5 6 7	ltrnel	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (⦋ G / g ⦌ X (P) \in A \land \neg ⦋ G / g ⦌ X (P) \underset{˙}{\leq} W)$
22	13 20 17 21	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (⦋ G / g ⦌ X (P) \in A \land \neg ⦋ G / g ⦌ X (P) \underset{˙}{\leq} W)$
23	22	simpld	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X (P) \in A$
24		simp31	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to I \in T$
25		simp32	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to I \neq {I ↾}_{B}$
26	1 5 6 7 8	trlnidat	$⊢ ((K \in HL \land W \in H) \land I \in T \land I \neq {I ↾}_{B}) \to R (I) \in A$
27	13 24 25 26	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (I) \in A$
28	24 25	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (I \in T \land I \neq {I ↾}_{B})$
29	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (I \in T \land I \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ I / g ⦌ X \in T$
30	13 14 28 16 17 18 29	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X \in T$
31		simp22l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to P \in A$
32	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land ⦋ I / g ⦌ X \in T \land P \in A) \to ⦋ I / g ⦌ X (P) \in A$
33	13 30 31 32	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X (P) \in A$
34		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to G \in T$
35		simp13r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to G \neq {I ↾}_{B}$
36	1 5 6 7 8	trlnidat	$⊢ ((K \in HL \land W \in H) \land G \in T \land G \neq {I ↾}_{B}) \to R (G) \in A$
37	13 34 35 36	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (G) \in A$
38	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land G \in T \land I \in T) \to G \circ I \in T$
39	13 34 24 38	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to G \circ I \in T$
40	34 24	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (G \in T \land I \in T)$
41		simp33	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (G) \neq R (I)$
42	1 6 7 8	trlconid	$⊢ ((K \in HL \land W \in H) \land (G \in T \land I \in T) \land R (G) \neq R (I)) \to G \circ I \neq {I ↾}_{B}$
43	13 40 41 42	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to G \circ I \neq {I ↾}_{B}$
44	39 43	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (G \circ I \in T \land G \circ I \neq {I ↾}_{B})$
45	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \circ I \in T \land G \circ I \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ (G \circ I) / g ⦌ X \in T$
46	13 14 44 16 17 18 45	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X \in T$
47	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land ⦋ (G \circ I) / g ⦌ X \in T \land P \in A) \to ⦋ (G \circ I) / g ⦌ X (P) \in A$
48	13 46 31 47	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X (P) \in A$
49	24 25 43	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})$
50	1 2 3 4 5 6 7 8 9 10 11	cdlemk46	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I))$
51	49 50	syld3an3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I))$
52	1 2 3 4 5 6 7 8 9 10 11	cdlemk45	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land G \circ I \neq {I ↾}_{B})) \to ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$
53	49 52	syld3an3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$
54	2 6 7 8	trlle	$⊢ ((K \in HL \land W \in H) \land I \in T) \to R (I) \underset{˙}{\leq} W$
55	13 24 54	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (I) \underset{˙}{\leq} W$
56	27 55	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (R (I) \in A \land R (I) \underset{˙}{\leq} W)$
57	2 6 7 8	trlle	$⊢ ((K \in HL \land W \in H) \land G \in T) \to R (G) \underset{˙}{\leq} W$
58	13 34 57	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (G) \underset{˙}{\leq} W$
59	37 58	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (R (G) \in A \land R (G) \underset{˙}{\leq} W)$
60	41	necomd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (I) \neq R (G)$
61	2 3 5 6	lhp2atne	$⊢ (((K \in HL \land W \in H) \land (⦋ G / g ⦌ X (P) \in A \land \neg ⦋ G / g ⦌ X (P) \underset{˙}{\leq} W) \land ⦋ I / g ⦌ X (P) \in A) \land ((R (I) \in A \land R (I) \underset{˙}{\leq} W) \land (R (G) \in A \land R (G) \underset{˙}{\leq} W)) \land R (I) \neq R (G)) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \neq ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)$
62	13 22 33 56 59 60 61	syl321anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \neq ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)$
63	2 3 4 5	2atm	$⊢ ((K \in HL \land ⦋ G / g ⦌ X (P) \in A \land R (I) \in A) \land (⦋ I / g ⦌ X (P) \in A \land R (G) \in A \land ⦋ (G \circ I) / g ⦌ X (P) \in A) \land (⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \land ⦋ (G \circ I) / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G)) \land ⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I) \neq ⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))) \to ⦋ (G \circ I) / g ⦌ X (P) = (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$
64	12 23 27 33 37 48 51 53 62 63	syl333anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X (P) = (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (I)) \underset{˙}{\land} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (G))$

Metamath Proof Explorer

Theorem cdlemk47

Proof