cdlemk19

Description: Part of proof of Lemma K of Crawley p. 118. Line 22 on p. 119. N , U , O , D are k, sigma_1 (p), k_1, f_1. (Contributed by NM, 2-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk1.b	$⊢ B = {Base}_{K}$
	cdlemk1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk1.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk1.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk1.a	$⊢ A = Atoms (K)$
	cdlemk1.h	$⊢ H = LHyp (K)$
	cdlemk1.t	$⊢ T = LTrn (K) (W)$
	cdlemk1.r	$⊢ R = trL (K) (W)$
	cdlemk1.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk1.o	$⊢ O = S (D)$
	cdlemk1.u	$⊢ U = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
Assertion	cdlemk19	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to U (F) = N$

Proof

Step	Hyp	Ref	Expression
1		cdlemk1.b	$⊢ B = {Base}_{K}$
2		cdlemk1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk1.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk1.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk1.a	$⊢ A = Atoms (K)$
6		cdlemk1.h	$⊢ H = LHyp (K)$
7		cdlemk1.t	$⊢ T = LTrn (K) (W)$
8		cdlemk1.r	$⊢ R = trL (K) (W)$
9		cdlemk1.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
10		cdlemk1.o	$⊢ O = S (D)$
11		cdlemk1.u	$⊢ U = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
12		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (K \in HL \land W \in H)$
13		simp21	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N \in T$
14		simp23	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to R (F) = R (N)$
15		simp12	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to F \in T$
16		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to D \in T$
17		simp33	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to R (D) \neq R (F)$
18	17 17	jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (R (D) \neq R (F) \land R (D) \neq R (F))$
19		simp31	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to F \neq {I ↾}_{B}$
20		simp32	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to D \neq {I ↾}_{B}$
21	19 19 20	3jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (F \neq {I ↾}_{B} \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B})$
22		simp22	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
23	1 2 3 4 5 6 7 8 9 10 11	cdlemkuel	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land F \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (F)) \land (F \neq {I ↾}_{B} \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (F) \in T$
24	12 14 15 15 16 13 18 21 22 23	syl333anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to U (F) \in T$
25	1 2 3 4 5 6 7 8 9 10 11	cdlemk18	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) = U (F) (P)$
26	2 5 6 7	cdlemd	$⊢ (((K \in HL \land W \in H) \land N \in T \land U (F) \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land N (P) = U (F) (P)) \to N = U (F)$
27	12 13 24 22 25 26	syl311anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N = U (F)$
28	27	eqcomd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to U (F) = N$

Metamath Proof Explorer

Theorem cdlemk19

Proof