cdlemk11ta

Description: Part of proof of Lemma K of Crawley p. 118. Lemma for Eq. 5, p. 119. G , I stand for g, h. TODO: fix comment. (Contributed by NM, 21-Jul-2013)

	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	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
	cdlemk5c.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk5a.u2	$⊢ C = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1}))))$
Assertion	cdlemk11ta	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to ⦋ G / g ⦌ Y \underset{˙}{\leq} (⦋ I / g ⦌ Y \underset{˙}{\lor} R (I \circ G^{-1}))$

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		cdlemk5c.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
12		cdlemk5a.u2	$⊢ C = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1}))))$
13		simp11	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (K \in HL \land W \in H)$
14		simp12l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to F \in T$
15		simp31	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to b \in T$
16		simp21	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to N \in T$
17		simp13l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to G \in T$
18		simp331	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to I \in T$
19	16 17 18	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (N \in T \land G \in T \land I \in T)$
20		simp22	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
21		simp23	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to R (F) = R (N)$
22		simp12r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to F \neq {I ↾}_{B}$
23		simp321	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to b \neq {I ↾}_{B}$
24		simp13r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to G \neq {I ↾}_{B}$
25	22 23 24	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B})$
26		simp332	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to I \neq {I ↾}_{B}$
27		simp322	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to R (b) \neq R (F)$
28		simp323	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to R (b) \neq R (G)$
29	28	necomd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to R (G) \neq R (b)$
30		simp333	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to R (b) \neq R (I)$
31	30	necomd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to R (I) \neq R (b)$
32	27 29 31	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (R (b) \neq R (F) \land R (G) \neq R (b) \land R (I) \neq R (b))$
33		eqid	$⊢ S (b) = S (b)$
34		eqid	$⊢ (G (P) \underset{˙}{\lor} I (P)) \underset{˙}{\land} (R (G \circ b^{-1}) \underset{˙}{\lor} R (I \circ b^{-1})) = (G (P) \underset{˙}{\lor} I (P)) \underset{˙}{\land} (R (G \circ b^{-1}) \underset{˙}{\lor} R (I \circ b^{-1}))$
35	1 2 3 4 5 6 7 8 11 33 12 34	cdlemk11u	$⊢ (((K \in HL \land W \in H) \land F \in T \land b \in T) \land ((N \in T \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land I \neq {I ↾}_{B} \land (R (b) \neq R (F) \land R (G) \neq R (b) \land R (I) \neq R (b)))) \to C (G) (P) \underset{˙}{\leq} (C (I) (P) \underset{˙}{\lor} R (I \circ G^{-1}))$
36	13 14 15 19 20 21 25 26 32 35	syl333anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to C (G) (P) \underset{˙}{\leq} (C (I) (P) \underset{˙}{\lor} R (I \circ G^{-1}))$
37		simp32	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G))$
38	15 37	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))$
39	1 2 3 4 5 6 7 8 9 10 11 12	cdlemkyuu	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ Y = C (G) (P)$
40	38 39	syld3an3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to ⦋ G / g ⦌ Y = C (G) (P)$
41		simp12	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (F \in T \land F \neq {I ↾}_{B})$
42	18 26	jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (I \in T \land I \neq {I ↾}_{B})$
43		simp2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))$
44	23 27 30	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (I))$
45	1 2 3 4 5 6 7 8 9 10 11 12	cdlemkyuu	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (I \in T \land I \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (I)))) \to ⦋ I / g ⦌ Y = C (I) (P)$
46	13 41 42 43 15 44 45	syl312anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to ⦋ I / g ⦌ Y = C (I) (P)$
47	46	oveq1d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to ⦋ I / g ⦌ Y \underset{˙}{\lor} R (I \circ G^{-1}) = C (I) (P) \underset{˙}{\lor} R (I \circ G^{-1})$
48	36 40 47	3brtr4d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \land (I \in T \land I \neq {I ↾}_{B} \land R (b) \neq R (I)))) \to ⦋ G / g ⦌ Y \underset{˙}{\leq} (⦋ I / g ⦌ Y \underset{˙}{\lor} R (I \circ G^{-1}))$

Metamath Proof Explorer

Theorem cdlemk11ta

Proof