cdlemk13-2N

Description: Part of proof of Lemma K of Crawley p. 118. Line 13 on p. 119. Q , C are k_2, f_2. (Contributed by NM, 1-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk2.b	$⊢ B = {Base}_{K}$
	cdlemk2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk2.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk2.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk2.a	$⊢ A = Atoms (K)$
	cdlemk2.h	$⊢ H = LHyp (K)$
	cdlemk2.t	$⊢ T = LTrn (K) (W)$
	cdlemk2.r	$⊢ R = trL (K) (W)$
	cdlemk2.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk2.q	$⊢ Q = S (C)$
Assertion	cdlemk13-2N	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to Q (P) = (P \underset{˙}{\lor} R (C)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (C \circ F^{-1}))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk2.b	$⊢ B = {Base}_{K}$
2		cdlemk2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk2.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk2.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk2.a	$⊢ A = Atoms (K)$
6		cdlemk2.h	$⊢ H = LHyp (K)$
7		cdlemk2.t	$⊢ T = LTrn (K) (W)$
8		cdlemk2.r	$⊢ R = trL (K) (W)$
9		cdlemk2.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		cdlemk2.q	$⊢ Q = S (C)$
11		simp11	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to K \in HL$
12		simp12	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to W \in H$
13	11 12	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
14		simp21	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \in T$
15		simp22	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to C \in T$
16		simp23	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to N \in T$
17		simp33	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18		simp13	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (F) = R (N)$
19		simp32l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \neq {I ↾}_{B}$
20		simp32r	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to C \neq {I ↾}_{B}$
21		simp31	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (C) \neq R (F)$
22	1 2 3 4 5 6 7 8 9 10	cdlemk13	$⊢ (((K \in HL \land W \in H) \land F \in T \land C \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land R (C) \neq R (F))) \to Q (P) = (P \underset{˙}{\lor} R (C)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (C \circ F^{-1}))$
23	13 14 15 16 17 18 19 20 21 22	syl333anc	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land (F \in T \land C \in T \land N \in T) \land (R (C) \neq R (F) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to Q (P) = (P \underset{˙}{\lor} R (C)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (C \circ F^{-1}))$

Metamath Proof Explorer

Theorem cdlemk13-2N

Proof