cdlemk55

Description: Part of proof of Lemma K of Crawley p. 118. Line 11, p. 120. G , I stand for g, h. X represents tau. (Contributed by NM, 26-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	cdlemk55	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

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		simpl1	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) = R (I)) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
13		simpl21	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) = R (I)) \to (F \in T \land F \neq {I ↾}_{B} \land N \in T)$
14		simpl22	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) = R (I)) \to G \in T$
15		simpl3	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) = R (I)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
16		simpl23	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) = R (I)) \to I \in T$
17		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) = R (I)) \to R (G) = R (I)$
18	1 2 3 4 5 6 7 8 9 10 11	cdlemk55b	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
19	12 13 14 15 16 17 18	syl132anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) = R (I)) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
20		simpl1	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) \neq R (I)) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
21		simpl21	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) \neq R (I)) \to (F \in T \land F \neq {I ↾}_{B} \land N \in T)$
22		simpl22	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) \neq R (I)) \to G \in T$
23		simpl3	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) \neq R (I)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
24		simpl23	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) \neq R (I)) \to I \in T$
25		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) \neq R (I)) \to R (G) \neq R (I)$
26	1 2 3 4 5 6 7 8 9 10 11	cdlemk53	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
27	20 21 22 23 24 25 26	syl132anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land R (G) \neq R (I)) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
28	19 27	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

Metamath Proof Explorer

Theorem cdlemk55

Proof