cdlemk14

Description: Part of proof of Lemma K of Crawley p. 118. Line 19 on p. 119. O , D are k_1, f_1. (Contributed by NM, 1-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	$⊢ O = S (D)$
Assertion	cdlemk14	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) \underset{˙}{\leq} (O (P) \underset{˙}{\lor} R (F \circ D^{-1}))$

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	1 2 3 4 5 6 7 8 9 10	cdlemk13	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to O (P) = (P \underset{˙}{\lor} R (D)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (D \circ F^{-1}))$
12		simp11l	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to K \in HL$
13	12	hllatd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to K \in Lat$
14		simp22l	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to P \in A$
15		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (K \in HL \land W \in H)$
16		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to D \in T$
17		simp32	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to D \neq {I ↾}_{B}$
18	1 5 6 7 8	trlnidat	$⊢ ((K \in HL \land W \in H) \land D \in T \land D \neq {I ↾}_{B}) \to R (D) \in A$
19	15 16 17 18	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to R (D) \in A$
20	1 3 5	hlatjcl	$⊢ (K \in HL \land P \in A \land R (D) \in A) \to P \underset{˙}{\lor} R (D) \in B$
21	12 14 19 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to P \underset{˙}{\lor} R (D) \in B$
22		simp21	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N \in T$
23	2 5 6 7	ltrnat	$⊢ ((K \in HL \land W \in H) \land N \in T \land P \in A) \to N (P) \in A$
24	15 22 14 23	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) \in A$
25		simp12	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to F \in T$
26		simp33	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to R (D) \neq R (F)$
27	5 6 7 8	trlcocnvat	$⊢ ((K \in HL \land W \in H) \land (D \in T \land F \in T) \land R (D) \neq R (F)) \to R (D \circ F^{-1}) \in A$
28	15 16 25 26 27	syl121anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to R (D \circ F^{-1}) \in A$
29	1 3 5	hlatjcl	$⊢ (K \in HL \land N (P) \in A \land R (D \circ F^{-1}) \in A) \to N (P) \underset{˙}{\lor} R (D \circ F^{-1}) \in B$
30	12 24 28 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) \underset{˙}{\lor} R (D \circ F^{-1}) \in B$
31	1 2 4	latmle2	$⊢ (K \in Lat \land P \underset{˙}{\lor} R (D) \in B \land N (P) \underset{˙}{\lor} R (D \circ F^{-1}) \in B) \to ((P \underset{˙}{\lor} R (D)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (D \circ F^{-1}))) \underset{˙}{\leq} (N (P) \underset{˙}{\lor} R (D \circ F^{-1}))$
32	13 21 30 31	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to ((P \underset{˙}{\lor} R (D)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (D \circ F^{-1}))) \underset{˙}{\leq} (N (P) \underset{˙}{\lor} R (D \circ F^{-1}))$
33	11 32	eqbrtrd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to O (P) \underset{˙}{\leq} (N (P) \underset{˙}{\lor} R (D \circ F^{-1}))$
34	10	fveq1i	$⊢ O (P) = S (D) (P)$
35	1 2 3 5 6 7 8 4 9	cdlemksat	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to S (D) (P) \in A$
36	34 35	eqeltrid	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to O (P) \in A$
37	6 7	ltrncnv	$⊢ ((K \in HL \land W \in H) \land F \in T) \to F^{-1} \in T$
38	15 25 37	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to F^{-1} \in T$
39	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land D \in T \land F^{-1} \in T) \to D \circ F^{-1} \in T$
40	15 16 38 39	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to D \circ F^{-1} \in T$
41	2 6 7 8	trlle	$⊢ ((K \in HL \land W \in H) \land D \circ F^{-1} \in T) \to R (D \circ F^{-1}) \underset{˙}{\leq} W$
42	15 40 41	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to R (D \circ F^{-1}) \underset{˙}{\leq} W$
43	1 2 3 4 5 6 7 8 9 10	cdlemkoatnle	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (O (P) \in A \land \neg O (P) \underset{˙}{\leq} W)$
44	43	simprd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to \neg O (P) \underset{˙}{\leq} W$
45		nbrne2	$⊢ (R (D \circ F^{-1}) \underset{˙}{\leq} W \land \neg O (P) \underset{˙}{\leq} W) \to R (D \circ F^{-1}) \neq O (P)$
46	42 44 45	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to R (D \circ F^{-1}) \neq O (P)$
47	46	necomd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to O (P) \neq R (D \circ F^{-1})$
48	2 3 5	hlatexch2	$⊢ (K \in HL \land (O (P) \in A \land N (P) \in A \land R (D \circ F^{-1}) \in A) \land O (P) \neq R (D \circ F^{-1})) \to (O (P) \underset{˙}{\leq} (N (P) \underset{˙}{\lor} R (D \circ F^{-1})) \to N (P) \underset{˙}{\leq} (O (P) \underset{˙}{\lor} R (D \circ F^{-1})))$
49	12 36 24 28 47 48	syl131anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to (O (P) \underset{˙}{\leq} (N (P) \underset{˙}{\lor} R (D \circ F^{-1})) \to N (P) \underset{˙}{\leq} (O (P) \underset{˙}{\lor} R (D \circ F^{-1})))$
50	33 49	mpd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) \underset{˙}{\leq} (O (P) \underset{˙}{\lor} R (D \circ F^{-1}))$
51	6 7 8	trlcocnv	$⊢ ((K \in HL \land W \in H) \land D \in T \land F \in T) \to R (D \circ F^{-1}) = R (F \circ D^{-1})$
52	15 16 25 51	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to R (D \circ F^{-1}) = R (F \circ D^{-1})$
53	52	oveq2d	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to O (P) \underset{˙}{\lor} R (D \circ F^{-1}) = O (P) \underset{˙}{\lor} R (F \circ D^{-1})$
54	50 53	breqtrd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land R (D) \neq R (F))) \to N (P) \underset{˙}{\leq} (O (P) \underset{˙}{\lor} R (F \circ D^{-1}))$

Metamath Proof Explorer

Theorem cdlemk14

Proof