cdlemk38

Description: Part of proof of Lemma K of Crawley p. 118. Line 31, p. 119. TODO: derive more directly with r19.23 ? (Contributed by NM, 19-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk4.b	$⊢ B = {Base}_{K}$
	cdlemk4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk4.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk4.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk4.a	$⊢ A = Atoms (K)$
	cdlemk4.h	$⊢ H = LHyp (K)$
	cdlemk4.t	$⊢ T = LTrn (K) (W)$
	cdlemk4.r	$⊢ R = trL (K) (W)$
	cdlemk4.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
	cdlemk4.y	$⊢ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
	cdlemk4.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	cdlemk38	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk4.b	$⊢ B = {Base}_{K}$
2		cdlemk4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk4.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk4.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk4.a	$⊢ A = Atoms (K)$
6		cdlemk4.h	$⊢ H = LHyp (K)$
7		cdlemk4.t	$⊢ T = LTrn (K) (W)$
8		cdlemk4.r	$⊢ R = trL (K) (W)$
9		cdlemk4.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
10		cdlemk4.y	$⊢ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
11		cdlemk4.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	1 6 7 8	cdlemftr2	$⊢ (K \in HL \land W \in H) \to \exists b \in T (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G))$
13	12	3ad2ant1	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to \exists b \in T (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G))$
14		nfv	$⊢ Ⅎ b ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)))$
15		nfra1	$⊢ Ⅎ b \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)$
16		nfcv	$⊢ \underline{Ⅎ} b T$
17	15 16	nfriota	$⊢ \underline{Ⅎ} b (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y))$
18	11 17	nfcxfr	$⊢ \underline{Ⅎ} b X$
19		nfcv	$⊢ \underline{Ⅎ} b P$
20	18 19	nffv	$⊢ \underline{Ⅎ} b X (P)$
21		nfcv	$⊢ \underline{Ⅎ} b \underset{˙}{\leq}$
22		nfcv	$⊢ \underline{Ⅎ} b (P \underset{˙}{\lor} R (G))$
23	20 21 22	nfbr	$⊢ Ⅎ b X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$
24		simpl1	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (K \in HL \land W \in H)$
25		simpl21	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \in T \land F \neq {I ↾}_{B})$
26		simpl22	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (G \in T \land G \neq {I ↾}_{B})$
27		simpl23	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to N \in T$
28		simpl3l	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
29		simpl3r	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (F) = R (N)$
30		simpr	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))$
31	1 2 3 4 5 6 7 8 9 10 11	cdlemk37	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$
32	24 25 26 27 28 29 30 31	syl331anc	$⊢ (((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$
33	32	exp32	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (b \in T \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))))$
34	14 23 33	rexlimd	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (\exists b \in T (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G)))$
35	13 34	mpd	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (G))$

Metamath Proof Explorer

Theorem cdlemk38

Proof