cdlemk26b-3

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 14-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	cdlemk26b-3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists x \in T ((x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)) \land x Y G \in T)$

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		simpl1	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
12	1 6 7 8	cdlemftr2	$⊢ (K \in HL \land W \in H) \to \exists x \in T (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G))$
13	11 12	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists x \in T (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G))$
14		simp3r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G))$
15		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to (K \in HL \land W \in H)$
16		simp133	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to R (F) = R (N)$
17		simp131	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to G \in T$
18		simp121	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to F \in T$
19		simp3l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to x \in T$
20		simp123	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to N \in T$
21		simp3r2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to R (x) \neq R (F)$
22		simp3r3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to R (x) \neq R (G)$
23	21 22	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to (R (x) \neq R (F) \land R (x) \neq R (G))$
24		simp122	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to F \neq {I ↾}_{B}$
25		simp132	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to G \neq {I ↾}_{B}$
26		simp3r1	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to x \neq {I ↾}_{B}$
27	24 25 26	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land x \neq {I ↾}_{B})$
28		simp2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
29	1 2 3 4 5 6 7 8 9 10	cdlemkuel-3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land x \in T \land N \in T) \land ((R (x) \neq R (F) \land R (x) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land x \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to x Y G \in T$
30	15 16 17 18 19 20 23 27 28 29	syl333anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to x Y G \in T$
31	14 30	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)))) \to ((x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)) \land x Y G \in T)$
32	31	3expia	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to ((x \in T \land (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G))) \to ((x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)) \land x Y G \in T))$
33	32	expd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (x \in T \to ((x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)) \to ((x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)) \land x Y G \in T)))$
34	33	reximdvai	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (\exists x \in T (x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)) \to \exists x \in T ((x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)) \land x Y G \in T))$
35	13 34	mpd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B} \land R (F) = R (N))) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists x \in T ((x \neq {I ↾}_{B} \land R (x) \neq R (F) \land R (x) \neq R (G)) \land x Y G \in T)$

Metamath Proof Explorer

Theorem cdlemk26b-3

Proof