cdlemk6

Description: Part of proof of Lemma K of Crawley p. 118. Apply dalaw . (Contributed by NM, 25-Jun-2013)

	Ref	Expression
Hypotheses	cdlemk.b	$⊢ B = {Base}_{K}$
	cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk.a	$⊢ A = Atoms (K)$
	cdlemk.h	$⊢ H = LHyp (K)$
	cdlemk.t	$⊢ T = LTrn (K) (W)$
	cdlemk.r	$⊢ R = trL (K) (W)$
	cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
Assertion	cdlemk6	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (((G (P) \underset{˙}{\lor} X (P)) \underset{˙}{\land} (R (G \circ F^{-1}) \underset{˙}{\lor} R (X \circ F^{-1}))) \underset{˙}{\lor} ((X (P) \underset{˙}{\lor} P) \underset{˙}{\land} (R (X \circ F^{-1}) \underset{˙}{\lor} N (P))))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk.b	$⊢ B = {Base}_{K}$
2		cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk.a	$⊢ A = Atoms (K)$
5		cdlemk.h	$⊢ H = LHyp (K)$
6		cdlemk.t	$⊢ T = LTrn (K) (W)$
7		cdlemk.r	$⊢ R = trL (K) (W)$
8		cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
9		simp31	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to F \neq {I ↾}_{B}$
10		simp32	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to G \neq {I ↾}_{B}$
11		simp33l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to R (G) \neq R (F)$
12	9 10 11	3jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))$
13	1 2 3 4 5 6 7 8	cdlemk5	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to ((P \underset{˙}{\lor} N (P)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$
14	12 13	syld3an3	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to ((P \underset{˙}{\lor} N (P)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$
15		simp11l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to K \in HL$
16		simp22l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to P \in A$
17		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to (K \in HL \land W \in H)$
18		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to G \in T$
19	2 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land G \in T \land P \in A) \to G (P) \in A$
20	17 18 16 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to G (P) \in A$
21		simp21r	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to X \in T$
22	2 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land X \in T \land P \in A) \to X (P) \in A$
23	17 21 16 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to X (P) \in A$
24		simp21l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to N \in T$
25	2 4 5 6	ltrnat	$⊢ ((K \in HL \land W \in H) \land N \in T \land P \in A) \to N (P) \in A$
26	17 24 16 25	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to N (P) \in A$
27		simp12	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to F \in T$
28	4 5 6 7	trlcocnvat	$⊢ ((K \in HL \land W \in H) \land (G \in T \land F \in T) \land R (G) \neq R (F)) \to R (G \circ F^{-1}) \in A$
29	17 18 27 11 28	syl121anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to R (G \circ F^{-1}) \in A$
30		simp33r	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to R (X) \neq R (F)$
31	4 5 6 7	trlcocnvat	$⊢ ((K \in HL \land W \in H) \land (X \in T \land F \in T) \land R (X) \neq R (F)) \to R (X \circ F^{-1}) \in A$
32	17 21 27 30 31	syl121anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to R (X \circ F^{-1}) \in A$
33	2 3 8 4	dalaw	$⊢ (K \in HL \land (P \in A \land G (P) \in A \land X (P) \in A) \land (N (P) \in A \land R (G \circ F^{-1}) \in A \land R (X \circ F^{-1}) \in A)) \to (((P \underset{˙}{\lor} N (P)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1})) \to ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (((G (P) \underset{˙}{\lor} X (P)) \underset{˙}{\land} (R (G \circ F^{-1}) \underset{˙}{\lor} R (X \circ F^{-1}))) \underset{˙}{\lor} ((X (P) \underset{˙}{\lor} P) \underset{˙}{\land} (R (X \circ F^{-1}) \underset{˙}{\lor} N (P)))))$
34	15 16 20 23 26 29 32 33	syl133anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to (((P \underset{˙}{\lor} N (P)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1})) \to ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (((G (P) \underset{˙}{\lor} X (P)) \underset{˙}{\land} (R (G \circ F^{-1}) \underset{˙}{\lor} R (X \circ F^{-1}))) \underset{˙}{\lor} ((X (P) \underset{˙}{\lor} P) \underset{˙}{\land} (R (X \circ F^{-1}) \underset{˙}{\lor} N (P)))))$
35	14 34	mpd	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land (R (G) \neq R (F) \land R (X) \neq R (F)))) \to ((P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (((G (P) \underset{˙}{\lor} X (P)) \underset{˙}{\land} (R (G \circ F^{-1}) \underset{˙}{\lor} R (X \circ F^{-1}))) \underset{˙}{\lor} ((X (P) \underset{˙}{\lor} P) \underset{˙}{\land} (R (X \circ F^{-1}) \underset{˙}{\lor} N (P))))$

Metamath Proof Explorer

Theorem cdlemk6

Proof