cdlemksv2

Description: Part of proof of Lemma K of Crawley p. 118. Value of the sigma(p) function S at the fixed P parameter. (Contributed by NM, 26-Jun-2013)

	Ref	Expression
Hypotheses	cdlemk.b	$⊢ B = {Base}_{K}$
	cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk.a	$⊢ A = Atoms (K)$
	cdlemk.h	$⊢ H = LHyp (K)$
	cdlemk.t	$⊢ T = LTrn (K) (W)$
	cdlemk.r	$⊢ R = trL (K) (W)$
	cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
Assertion	cdlemksv2	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to S (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk.b	$⊢ B = {Base}_{K}$
2		cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk.a	$⊢ A = Atoms (K)$
5		cdlemk.h	$⊢ H = LHyp (K)$
6		cdlemk.t	$⊢ T = LTrn (K) (W)$
7		cdlemk.r	$⊢ R = trL (K) (W)$
8		cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
9		cdlemk.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		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to G \in T$
11	1 2 3 4 5 6 7 8 9	cdlemksv	$⊢ G \in T \to S (G) = (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})))$
12	10 11	syl	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to S (G) = (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})))$
13	12	eqcomd	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) = S (G)$
14	1 2 3 4 5 6 7 8 9	cdlemksel	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to S (G) \in T$
15		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (K \in HL \land W \in H)$
16		simp22	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
17		simp1	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to ((K \in HL \land W \in H) \land F \in T \land G \in T)$
18		simp21	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N \in T$
19	2 4 5 6	ltrnel	$⊢ ((K \in HL \land W \in H) \land N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (N (P) \in A \land \neg N (P) \underset{˙}{\leq} W)$
20	15 18 16 19	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (N (P) \in A \land \neg N (P) \underset{˙}{\leq} W)$
21		simp11l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to K \in HL$
22		simp22l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \in A$
23	20	simpld	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N (P) \in A$
24	2 3 4	hlatlej2	$⊢ (K \in HL \land P \in A \land N (P) \in A) \to N (P) \underset{˙}{\leq} (P \underset{˙}{\lor} N (P))$
25	21 22 23 24	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N (P) \underset{˙}{\leq} (P \underset{˙}{\lor} N (P))$
26		simp23	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to R (F) = R (N)$
27	26	oveq2d	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \underset{˙}{\lor} R (F) = P \underset{˙}{\lor} R (N)$
28	2 3 4 5 6 7	trljat1	$⊢ ((K \in HL \land W \in H) \land N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to P \underset{˙}{\lor} R (N) = P \underset{˙}{\lor} N (P)$
29	15 18 16 28	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \underset{˙}{\lor} R (N) = P \underset{˙}{\lor} N (P)$
30	27 29	eqtr2d	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \underset{˙}{\lor} N (P) = P \underset{˙}{\lor} R (F)$
31	25 30	breqtrd	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (F))$
32		simp31	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to F \neq {I ↾}_{B}$
33		simp32	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to G \neq {I ↾}_{B}$
34		simp33	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to R (G) \neq R (F)$
35	34	necomd	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to R (F) \neq R (G)$
36		eqid	$⊢ (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))$
37	1 2 3 8 4 5 6 7 36	cdlemh	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (N (P) \in A \land \neg N (P) \underset{˙}{\leq} W) \land N (P) \underset{˙}{\leq} (P \underset{˙}{\lor} R (F))) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (F) \neq R (G))) \to ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} W)$
38	17 16 20 31 32 33 35 37	syl133anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} W)$
39	2 4 5 6	cdleme	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \in A \land \neg ((P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} W)) \to \exists! i \in T i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))$
40	15 16 38 39	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to \exists! i \in T i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))$
41		nfcv	$⊢ \underline{Ⅎ} i T$
42		nfriota1	$⊢ \underline{Ⅎ} i (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1})))$
43	41 42	nfmpt	$⊢ \underline{Ⅎ} i (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
44	9 43	nfcxfr	$⊢ \underline{Ⅎ} i S$
45		nfcv	$⊢ \underline{Ⅎ} i G$
46	44 45	nffv	$⊢ \underline{Ⅎ} i S (G)$
47		nfcv	$⊢ \underline{Ⅎ} i P$
48	46 47	nffv	$⊢ \underline{Ⅎ} i S (G) (P)$
49	48	nfeq1	$⊢ Ⅎ i S (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))$
50		fveq1	$⊢ i = S (G) \to i (P) = S (G) (P)$
51	50	eqeq1d	$⊢ i = S (G) \to (i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \leftrightarrow S (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})))$
52	46 49 51	riota2f	$⊢ (S (G) \in T \land \exists! i \in T i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \to (S (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \leftrightarrow (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) = S (G))$
53	14 40 52	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (S (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1})) \leftrightarrow (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))) = S (G))$
54	13 53	mpbird	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to S (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (G \circ F^{-1}))$

Metamath Proof Explorer

Theorem cdlemksv2

Proof