Metamath Proof Explorer

Theorem cdlemk42yN

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. (Contributed by NM, 20-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	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
		cdlemk5.y	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
		cdlemk5.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
	Assertion	cdlemk42yN	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$

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		cdlemk5.y	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
11		cdlemk5.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
12	1 2 3 4 5 6 7 8 9 10 11	cdlemk42	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y$
13		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \in T$
14	10	cdlemk41	$⊢ G \in T \to ⦋ G / g ⦌ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
15	13 14	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
16	12 15	eqtrd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$