cdlemk27-3

Description: Part of proof of Lemma K of Crawley p. 118. Eliminate the P from the conclusion of cdlemk25-3 . (Contributed by NM, 10-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk3.b	$⊢ B = {Base}_{K}$
	cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk3.a	$⊢ A = Atoms (K)$
	cdlemk3.h	$⊢ H = LHyp (K)$
	cdlemk3.t	$⊢ T = LTrn (K) (W)$
	cdlemk3.r	$⊢ R = trL (K) (W)$
	cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
Assertion	cdlemk27-3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to D Y G = C Y G$

Proof

Step	Hyp	Ref	Expression
1		cdlemk3.b	$⊢ B = {Base}_{K}$
2		cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk3.a	$⊢ A = Atoms (K)$
6		cdlemk3.h	$⊢ H = LHyp (K)$
7		cdlemk3.t	$⊢ T = LTrn (K) (W)$
8		cdlemk3.r	$⊢ R = trL (K) (W)$
9		cdlemk3.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		cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
11		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (K \in HL \land W \in H)$
12		simp221	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to R (F) = R (N)$
13		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to G \in T$
14		simp12	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (F \in T \land D \in T \land N \in T)$
15		simp3l3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to R (D) \neq R (F)$
16		simp3r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to R (G) \neq R (D)$
17	16	necomd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to R (D) \neq R (G)$
18	15 17	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (R (D) \neq R (F) \land R (D) \neq R (G))$
19		simp222	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to F \neq {I ↾}_{B}$
20		simp23l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to G \neq {I ↾}_{B}$
21		simp223	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to D \neq {I ↾}_{B}$
22	19 20 21	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B})$
23		simp21	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
24	1 2 3 4 5 6 7 8 9 10	cdlemkuel-3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to D Y G \in T$
25	11 12 13 14 18 22 23 24	syl313anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to D Y G \in T$
26		simp121	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to F \in T$
27		simp13r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to C \in T$
28		simp123	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to N \in T$
29		simp3l2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to R (C) \neq R (F)$
30		simp3l1	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to R (G) \neq R (C)$
31	30	necomd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to R (C) \neq R (G)$
32	29 31	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (R (C) \neq R (F) \land R (C) \neq R (G))$
33		simp23r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to C \neq {I ↾}_{B}$
34	19 20 33	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})$
35	1 2 3 4 5 6 7 8 9 10	cdlemkuel-3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land C \in T \land N \in T) \land ((R (C) \neq R (F) \land R (C) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to C Y G \in T$
36	11 12 13 26 27 28 32 34 23 35	syl333anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to C Y G \in T$
37	1 2 3 4 5 6 7 8 9 10	cdlemk26-3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to (D Y G) (P) = (C Y G) (P)$
38	2 5 6 7	cdlemd	$⊢ (((K \in HL \land W \in H) \land D Y G \in T \land C Y G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (D Y G) (P) = (C Y G) (P)) \to D Y G = C Y G$
39	11 25 36 23 37 38	syl311anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land D \in T \land N \in T) \land (G \in T \land C \in T)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (R (F) = R (N) \land F \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (G \neq {I ↾}_{B} \land C \neq {I ↾}_{B})) \land ((R (G) \neq R (C) \land R (C) \neq R (F) \land R (D) \neq R (F)) \land R (G) \neq R (D))) \to D Y G = C Y G$

Metamath Proof Explorer

Theorem cdlemk27-3

Proof