cdlemk56w

Description: Use a fixed element to eliminate P in cdlemk56 . (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	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
	cdlemk6.e	$⊢ E = TEndo (K) (W)$
Assertion	cdlemk56w	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to (U \in E \land U (F) = N)$

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		cdlemk6.e	$⊢ E = TEndo (K) (W)$
15		simp1	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to (K \in HL \land W \in H)$
16		simp2l	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to F \in T$
17		simp2r	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to N \in T$
18		simp3	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to R (F) = R (N)$
19		eqid	$⊢ \leq_{K} = \leq_{K}$
20	4	fveq1i	$⊢ \underset{˙}{⊥} (W) = oc (K) (W)$
21	9 20	eqtri	$⊢ P = oc (K) (W)$
22	19 5 6 21	lhpocnel2	$⊢ (K \in HL \land W \in H) \to (P \in A \land \neg P \leq_{K} W)$
23	22	3ad2ant1	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to (P \in A \land \neg P \leq_{K} W)$
24	1 19 2 3 5 6 7 8 10 11 12 13 14	cdlemk56	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \leq_{K} W)) \to U \in E$
25	15 16 17 18 23 24	syl311anc	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to U \in E$
26	1 2 3 4 5 6 7 8 9 10 11 12 13	cdlemk19w	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to U (F) = N$
27	25 26	jca	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land R (F) = R (N)) \to (U \in E \land U (F) = N)$

Metamath Proof Explorer

Theorem cdlemk56w

Proof