cdlemk49

Description: Part of proof of Lemma K of Crawley p. 118. Line 5, 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	cdlemk49	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X))$

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		simp13	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (G \in T \land G \neq {I ↾}_{B})$
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	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ G / g ⦌ X \in T$
19	12 13 14 15 16 17 18	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X \in T$
20		simp3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (I \in T \land I \neq {I ↾}_{B})$
21	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (I \in T \land I \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ I / g ⦌ X \in T$
22	12 13 20 15 16 17 21	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ I / g ⦌ X \in T$
23	6 7	ltrncom	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T \land ⦋ I / g ⦌ X \in T) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X = ⦋ I / g ⦌ X \circ ⦋ G / g ⦌ X$
24	12 19 22 23	syl3anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X = ⦋ I / g ⦌ X \circ ⦋ G / g ⦌ X$
25	24	fveq1d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) = (⦋ I / g ⦌ X \circ ⦋ G / g ⦌ X) (P)$
26		simp2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))$
27	1 2 3 4 5 6 7 8 9 10 11	cdlemk48	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (I \in T \land I \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (G \in T \land G \neq {I ↾}_{B})) \to (⦋ I / g ⦌ X \circ ⦋ G / g ⦌ X) (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X))$
28	12 13 20 26 14 27	syl311anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ I / g ⦌ X \circ ⦋ G / g ⦌ X) (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X))$
29	25 28	eqbrtrd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) (P) \underset{˙}{\leq} (⦋ G / g ⦌ X (P) \underset{˙}{\lor} R (⦋ I / g ⦌ X))$

Metamath Proof Explorer

Theorem cdlemk49

Proof