cdlemk11t

Description: Part of proof of Lemma K of Crawley p. 118. Eq. 5, line 36, p. 119. G , I stand for g, h. X represents tau. (Contributed by NM, 21-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	cdlemk11t	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ G^{-1}))$

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	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to K \in HL$
13		simp11r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to W \in H$
14	1 6 7 8	cdlemftr3	$⊢ (K \in HL \land W \in H) \to \exists b \in T (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))$
15	12 13 14	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to \exists b \in T (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))$
16		nfv	$⊢ Ⅎ b (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B}))$
17		nfcv	$⊢ \underline{Ⅎ} b G$
18		nfra1	$⊢ Ⅎ b \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y)$
19		nfcv	$⊢ \underline{Ⅎ} b T$
20	18 19	nfriota	$⊢ \underline{Ⅎ} b (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
21	11 20	nfcxfr	$⊢ \underline{Ⅎ} b X$
22	17 21	nfcsbw	$⊢ \underline{Ⅎ} b ⦋ G / g ⦌ X$
23		nfcv	$⊢ \underline{Ⅎ} b P$
24	22 23	nffv	$⊢ \underline{Ⅎ} b ⦋ G / g ⦌ X (P)$
25		nfcv	$⊢ \underline{Ⅎ} b \underset{˙}{\leq}$
26		nfcv	$⊢ \underline{Ⅎ} b I$
27	26 21	nfcsbw	$⊢ \underline{Ⅎ} b ⦋ I / g ⦌ X$
28	27 23	nffv	$⊢ \underline{Ⅎ} b ⦋ I / g ⦌ X (P)$
29		nfcv	$⊢ \underline{Ⅎ} b \underset{˙}{\lor}$
30		nfcv	$⊢ \underline{Ⅎ} b R (I \circ G^{-1})$
31	28 29 30	nfov	$⊢ \underline{Ⅎ} b (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ G^{-1}))$
32	24 25 31	nfbr	$⊢ Ⅎ b ⦋ G / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ G^{-1}))$
33		simp11	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to ((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}))$
34		simp12	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))$
35		simp2	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to b \in T$
36		simp3l	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to b \neq {I ↾}_{B}$
37		simp3r1	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to R (b) \neq R (F)$
38		simp3r2	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to R (b) \neq R (G)$
39	36 37 38	3jca	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G))$
40		simp13l	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to I \in T$
41		simp13r	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to I \neq {I ↾}_{B}$
42		simp3r3	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to R (b) \neq R (I)$
43	40 41 42	3jca	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I))$
44	1 2 3 4 5 6 7 8 9 10 11	cdlemk11tc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to ⦋ G / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ G^{-1}))$
45	33 34 35 39 43 44	syl113anc	$⊢ ((((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \land b \in T \land (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I)))) \to ⦋ G / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ G^{-1}))$
46	45	3exp	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (b \in T \to ((b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I))) \to ⦋ G / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ G^{-1}))))$
47	16 32 46	rexlimd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to (\exists b \in T (b \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (b) \neq R (G) \land R (b) \neq R (I))) \to ⦋ G / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ G^{-1})))$
48	15 47	mpd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B})) \to ⦋ G / g ⦌ X (P) \underset{˙}{\leq} (⦋ I / g ⦌ X (P) \underset{˙}{\lor} R (I \circ G^{-1}))$

Metamath Proof Explorer

Theorem cdlemk11t

Proof