cdlemk12u-2N

Description: Part of proof of Lemma K of Crawley p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma_2 ( V ) 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	$⊢ V = (d \in T ⟼ (ι k \in T \| k (P) = (P \underset{˙}{\lor} R (d)) \underset{˙}{\land} (Q (P) \underset{˙}{\lor} R (d \circ C^{-1}))))$
Assertion	cdlemk12u-2N	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to V (G) (P) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (V (X) (P) \underset{˙}{\lor} R (X \circ G^{-1}))$

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		simp11	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to K \in HL$
13		simp12	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to W \in H$
14	12 13	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
15		simp211	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \in T$
16		simp212	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to C \in T$
17		simp213	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to N \in T$
18		simp22l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \in T$
19		simp23l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to X \in T$
20	17 18 19	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (N \in T \land G \in T \land X \in T)$
21		simp33	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
22		simp13	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (F) = R (N)$
23		simp322	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \neq {I ↾}_{B}$
24		simp323	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to C \neq {I ↾}_{B}$
25		simp22r	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \neq {I ↾}_{B}$
26	23 24 25	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land G \neq {I ↾}_{B})$
27		simp23r	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to X \neq {I ↾}_{B}$
28		simp321	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (G) \neq R (X)$
29	27 28	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (X \neq {I ↾}_{B} \land R (G) \neq R (X))$
30		simp31	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C))$
31	1 2 3 4 5 6 7 8 9 10 11	cdlemk12u	$⊢ (((K \in HL \land W \in H) \land F \in T \land C \in T) \land ((N \in T \land G \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (X \neq {I ↾}_{B} \land R (G) \neq R (X)) \land (R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)))) \to V (G) (P) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (V (X) (P) \underset{˙}{\lor} R (X \circ G^{-1}))$
32	14 15 16 20 21 22 26 29 30 31	syl333anc	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land C \in T \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (X \in T \land X \neq {I ↾}_{B})) \land ((R (C) \neq R (F) \land R (G) \neq R (C) \land R (X) \neq R (C)) \land (R (G) \neq R (X) \land F \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to V (G) (P) = (P \underset{˙}{\lor} G (P)) \underset{˙}{\land} (V (X) (P) \underset{˙}{\lor} R (X \circ G^{-1}))$

Metamath Proof Explorer

Theorem cdlemk12u-2N

Proof