cdlemk19w

Description: Use a fixed element to eliminate P in cdlemk19u . (Contributed by NM, 1-Aug-2013)

	Ref	Expression
Hypotheses	cdlemk6.b	$⊢ B = {Base}_{K}$
	cdlemk6.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk6.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk6.o	$⊢ \underset{˙}{⊥} = oc (K)$
	cdlemk6.a	$⊢ A = Atoms (K)$
	cdlemk6.h	$⊢ H = LHyp (K)$
	cdlemk6.t	$⊢ T = LTrn (K) (W)$
	cdlemk6.r	$⊢ R = trL (K) (W)$
	cdlemk6.p	$⊢ P = \underset{˙}{⊥} (W)$
	cdlemk6.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
	cdlemk6.y	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
	cdlemk6.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
	cdlemk6.u	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
Assertion	cdlemk19w	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to U (F) = N$

Proof

Step	Hyp	Ref	Expression
1		cdlemk6.b	$⊢ B = {Base}_{K}$
2		cdlemk6.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemk6.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemk6.o	$⊢ \underset{˙}{⊥} = oc (K)$
5		cdlemk6.a	$⊢ A = Atoms (K)$
6		cdlemk6.h	$⊢ H = LHyp (K)$
7		cdlemk6.t	$⊢ T = LTrn (K) (W)$
8		cdlemk6.r	$⊢ R = trL (K) (W)$
9		cdlemk6.p	$⊢ P = \underset{˙}{⊥} (W)$
10		cdlemk6.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
11		cdlemk6.y	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
12		cdlemk6.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
13		cdlemk6.u	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
14		3simpb	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
15		simp2	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to (F \in T \land N \in T)$
16		eqid	$⊢ \leq_{K} = \leq_{K}$
17	16 4 5 6	lhpocnel	$⊢ (K \in HL \land W \in H) \to (\underset{˙}{⊥} (W) \in A \land \neg \underset{˙}{⊥} (W) \leq_{K} W)$
18	17	3ad2ant1	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to (\underset{˙}{⊥} (W) \in A \land \neg \underset{˙}{⊥} (W) \leq_{K} W)$
19	9	eleq1i	$⊢ P \in A \leftrightarrow \underset{˙}{⊥} (W) \in A$
20	9	breq1i	$⊢ P \leq_{K} W \leftrightarrow \underset{˙}{⊥} (W) \leq_{K} W$
21	20	notbii	$⊢ \neg P \leq_{K} W \leftrightarrow \neg \underset{˙}{⊥} (W) \leq_{K} W$
22	19 21	anbi12i	$⊢ (P \in A \land \neg P \leq_{K} W) \leftrightarrow (\underset{˙}{⊥} (W) \in A \land \neg \underset{˙}{⊥} (W) \leq_{K} W)$
23	18 22	sylibr	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to (P \in A \land \neg P \leq_{K} W)$
24	1 16 2 3 5 6 7 8 10 11 12 13	cdlemk19u	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T) \land (P \in A \land \neg P \leq_{K} W)) \to U (F) = N$
25	14 15 23 24	syl3anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to U (F) = N$

Metamath Proof Explorer

Theorem cdlemk19w

Proof