cdlemk30

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk3.b	$⊢ B = {Base}_{K}$
	cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk3.a	$⊢ A = Atoms (K)$
	cdlemk3.h	$⊢ H = LHyp (K)$
	cdlemk3.t	$⊢ T = LTrn (K) (W)$
	cdlemk3.r	$⊢ R = trL (K) (W)$
	cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
Assertion	cdlemk30	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to S (b) (P) = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk3.b	$⊢ B = {Base}_{K}$
2		cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk3.a	$⊢ A = Atoms (K)$
6		cdlemk3.h	$⊢ H = LHyp (K)$
7		cdlemk3.t	$⊢ T = LTrn (K) (W)$
8		cdlemk3.r	$⊢ R = trL (K) (W)$
9		cdlemk3.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		simp1l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
11		simp21	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \in T$
12		simp22	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b \in T$
13		simp23	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to N \in T$
14		simp33	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
15		simp1r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (F) = R (N)$
16		simp32l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \neq {I ↾}_{B}$
17		simp32r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b \neq {I ↾}_{B}$
18		simp31	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (b) \neq R (F)$
19	1 2 3 5 6 7 8 4 9	cdlemksv2	$⊢ (((K \in HL \land W \in H) \land F \in T \land b \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land R (b) \neq R (F))) \to S (b) (P) = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
20	10 11 12 13 14 15 16 17 18 19	syl333anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to S (b) (P) = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$

Metamath Proof Explorer

Theorem cdlemk30

Proof