Metamath Proof Explorer

Theorem cdlemkfid2N

Description: Lemma for cdlemkfid3N . (Contributed by NM, 29-Jul-2013) (New usage is discouraged.)

		Ref	Expression
	Hypotheses	cdlemk5.b	$⊢ B = {Base}_{K}$
		cdlemk5.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		cdlemk5.j	$⊢ \underset{˙}{\lor} = join (K)$
		cdlemk5.m	$⊢ \underset{˙}{\land} = meet (K)$
		cdlemk5.a	$⊢ A = Atoms (K)$
		cdlemk5.h	$⊢ H = LHyp (K)$
		cdlemk5.t	$⊢ T = LTrn (K) (W)$
		cdlemk5.r	$⊢ R = trL (K) (W)$
		cdlemk5.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
	Assertion	cdlemkfid2N	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to Z = b (P)$

Proof

Step	Hyp	Ref	Expression
1		cdlemk5.b	$⊢ B = {Base}_{K}$
2		cdlemk5.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk5.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk5.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk5.a	$⊢ A = Atoms (K)$
6		cdlemk5.h	$⊢ H = LHyp (K)$
7		cdlemk5.t	$⊢ T = LTrn (K) (W)$
8		cdlemk5.r	$⊢ R = trL (K) (W)$
9		cdlemk5.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
10		simp1r	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F = N$
11	10	fveq1d	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F (P) = N (P)$
12	11	oveq1d	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F (P) \underset{˙}{\lor} R (b \circ F^{-1}) = N (P) \underset{˙}{\lor} R (b \circ F^{-1})$
13	12	oveq2d	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (b \circ F^{-1})) = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
14	1 2 3 4 5 6 7 8	cdlemkfid1N	$⊢ ((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (b \circ F^{-1})) = b (P)$
15	14	3adant1r	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (b \circ F^{-1})) = b (P)$
16	13 15	eqtr3d	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1})) = b (P)$
17	9 16	eqtrid	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to Z = b (P)$