cdlemkyyN

Description: Part of proof of Lemma K of Crawley p. 118. TODO: clean up ( b Y G ) stuff. (Contributed by NM, 21-Jul-2013) (New usage is discouraged.)

	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	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
	cdlemk5.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
	cdlemk5a.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.u1	$⊢ V = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
Assertion	cdlemkyyN	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = (b V G) (P)$

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		cdlemk5.x	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
12		cdlemk5a.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
13		cdlemk5a.u1	$⊢ V = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
14		simp11	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to K \in HL$
15		simp12	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to W \in H$
16	14 15	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (K \in HL \land W \in H)$
17		simp13	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (F) = R (N)$
18		simp211	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \in T$
19		simp3l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to b \in T$
20		simp213	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to N \in T$
21		simp3r2	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (b) \neq R (F)$
22		simp212	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \neq {I ↾}_{B}$
23		simp3r1	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to b \neq {I ↾}_{B}$
24	22 23	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B})$
25		simp23	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
26	1 2 3 4 5 6 7 8 12	cdlemk30	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land b \in T \land N \in T) \land (R (b) \neq R (F) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to S (b) (P) = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
27	16 17 18 19 20 21 24 25 26	syl233anc	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to S (b) (P) = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
28	27 9	eqtr4di	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to S (b) (P) = Z$
29	28	oveq1d	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to S (b) (P) \underset{˙}{\lor} R (G \circ b^{-1}) = Z \underset{˙}{\lor} R (G \circ b^{-1})$
30	29	oveq2d	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (G \circ b^{-1})) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
31	18 19 20	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \in T \land b \in T \land N \in T)$
32		simp22l	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \in T$
33		simp3r3	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (b) \neq R (G)$
34	21 33	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (R (b) \neq R (F) \land R (b) \neq R (G))$
35		simp22r	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \neq {I ↾}_{B}$
36	22 23 35	3jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B})$
37	1 2 3 4 5 6 7 8 12 13	cdlemk31	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (b V G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (G \circ b^{-1}))$
38	16 17 31 32 34 36 25 37	syl223anc	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (b V G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (G \circ b^{-1}))$
39	18 22	jca	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \in T \land F \neq {I ↾}_{B})$
40		simp22	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (G \in T \land G \neq {I ↾}_{B})$
41		simp3	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))$
42	1 2 3 4 5 6 7 8 9 10 11	cdlemk42yN	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
43	16 39 40 20 25 17 41 42	syl331anc	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
44	30 38 43	3eqtr4rd	$⊢ ((K \in HL \land W \in H \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = (b V G) (P)$

Metamath Proof Explorer

Theorem cdlemkyyN

Proof