cdlemk7u-2N

Description: Part of proof of Lemma K of Crawley p. 118. Line 5, p. 119 for the sigma_2 case. (Contributed by NM, 5-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk2.b	$⊢ B = {Base}_{K}$
	cdlemk2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk2.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk2.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk2.a	$⊢ A = Atoms (K)$
	cdlemk2.h	$⊢ H = LHyp (K)$
	cdlemk2.t	$⊢ T = LTrn (K) (W)$
	cdlemk2.r	$⊢ R = trL (K) (W)$
	cdlemk2.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk2.q	$⊢ Q = S (C)$
	cdlemk2.v	$⊢ V = (d \in T ⟼ (ι k \in T \| k (P) = (P \underset{˙}{\lor} R (d)) \underset{˙}{\land} (Q (P) \underset{˙}{\lor} R (d \circ C^{-1}))))$
	cdlemk2.z	$⊢ Z = (G (P) \underset{˙}{\lor} X (P)) \underset{˙}{\land} (R (G \circ C^{-1}) \underset{˙}{\lor} R (X \circ C^{-1}))$
Assertion	cdlemk7u-2N	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to V (G) (P) \underset{˙}{\leq} (V (X) (P) \underset{˙}{\lor} Z)$

Proof

Step	Hyp	Ref	Expression
1		cdlemk2.b	$⊢ B = {Base}_{K}$
2		cdlemk2.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk2.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk2.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk2.a	$⊢ A = Atoms (K)$
6		cdlemk2.h	$⊢ H = LHyp (K)$
7		cdlemk2.t	$⊢ T = LTrn (K) (W)$
8		cdlemk2.r	$⊢ R = trL (K) (W)$
9		cdlemk2.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		cdlemk2.q	$⊢ Q = S (C)$
11		cdlemk2.v	$⊢ V = (d \in T ⟼ (ι k \in T \| k (P) = (P \underset{˙}{\lor} R (d)) \underset{˙}{\land} (Q (P) \underset{˙}{\lor} R (d \circ C^{-1}))))$
12		cdlemk2.z	$⊢ Z = (G (P) \underset{˙}{\lor} X (P)) \underset{˙}{\land} (R (G \circ C^{-1}) \underset{˙}{\lor} R (X \circ C^{-1}))$
13		simp11	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to K \in HL$
14		simp12	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to W \in H$
15	13 14	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
16		simp211	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \in T$
17		simp212	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to C \in T$
18		simp213	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to N \in T$
19		simp22l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \in T$
20		simp23l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to X \in T$
21	18 19 20	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (N \in T \land G \in T \land X \in T)$
22		simp33	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
23		simp13	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (F) = R (N)$
24		simp32l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \neq {I ↾}_{B}$
25		simp32r	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to C \neq {I ↾}_{B}$
26		simp22r	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \neq {I ↾}_{B}$
27	24 25 26	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land G \neq {I ↾}_{B})$
28		simp23r	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to X \neq {I ↾}_{B}$
29		simp31	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C))$
30	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk7u	$⊢ (((K \in HL \land W \in H) \land F \in T \land C \in T) \land ((N \in T \land G \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land X \neq {I ↾}_{B} \land (R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)))) \to V (G) (P) \underset{˙}{\leq} (V (X) (P) \underset{˙}{\lor} Z)$
31	15 16 17 21 22 23 27 28 29 30	syl333anc	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to V (G) (P) \underset{˙}{\leq} (V (X) (P) \underset{˙}{\lor} Z)$

Metamath Proof Explorer

Theorem cdlemk7u-2N

Proof