cdlemkuv2

Description: Part of proof of Lemma K of Crawley p. 118. Line 16 on p. 119 for i = 1, where sigma_1 (p) is U , f_1 is D , and k_1 is O . (Contributed by NM, 2-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk1.b	$⊢ B = {Base}_{K}$
	cdlemk1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk1.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk1.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk1.a	$⊢ A = Atoms (K)$
	cdlemk1.h	$⊢ H = LHyp (K)$
	cdlemk1.t	$⊢ T = LTrn (K) (W)$
	cdlemk1.r	$⊢ R = trL (K) (W)$
	cdlemk1.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk1.o	$⊢ O = S (D)$
	cdlemk1.u	$⊢ U = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
Assertion	cdlemkuv2	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk1.b	$⊢ B = {Base}_{K}$
2		cdlemk1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk1.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk1.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk1.a	$⊢ A = Atoms (K)$
6		cdlemk1.h	$⊢ H = LHyp (K)$
7		cdlemk1.t	$⊢ T = LTrn (K) (W)$
8		cdlemk1.r	$⊢ R = trL (K) (W)$
9		cdlemk1.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		cdlemk1.o	$⊢ O = S (D)$
11		cdlemk1.u	$⊢ U = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
12		simp13	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \in T$
13	1 2 3 5 6 7 8 4 11	cdlemksv	$⊢ G \in T \to U (G) = (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})))$
14	12 13	syl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (G) = (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})))$
15	14	eqcomd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))) = U (G)$
16	1 2 3 4 5 6 7 8 9 10 11	cdlemkuel	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (G) \in T$
17		simp11l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to K \in HL$
18		simp11r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to W \in H$
19		simp33	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
20	1 2 3 4 5 6 7 8 9 10	cdlemk16a	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))) \underset{˙}{\leq} W)$
21	2 5 6 7	cdleme	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))) \underset{˙}{\leq} W)) \to \exists! j \in T j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))$
22	17 18 19 20 21	syl211anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to \exists! j \in T j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))$
23		nfcv	$⊢ \underline{Ⅎ} j T$
24		nfriota1	$⊢ \underline{Ⅎ} j (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1})))$
25	23 24	nfmpt	$⊢ \underline{Ⅎ} j (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
26	11 25	nfcxfr	$⊢ \underline{Ⅎ} j U$
27		nfcv	$⊢ \underline{Ⅎ} j G$
28	26 27	nffv	$⊢ \underline{Ⅎ} j U (G)$
29		nfcv	$⊢ \underline{Ⅎ} j P$
30	28 29	nffv	$⊢ \underline{Ⅎ} j U (G) (P)$
31	30	nfeq1	$⊢ Ⅎ j U (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))$
32		fveq1	$⊢ j = U (G) \to j (P) = U (G) (P)$
33	32	eqeq1d	$⊢ j = U (G) \to (j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})) \leftrightarrow U (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})))$
34	28 31 33	riota2f	$⊢ (U (G) \in T \land \exists! j \in T j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))) \to (U (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})) \leftrightarrow (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))) = U (G))$
35	16 22 34	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (U (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1})) \leftrightarrow (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))) = U (G))$
36	15 35	mpbird	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (G \circ D^{-1}))$

Metamath Proof Explorer

Theorem cdlemkuv2

Proof