cdlemk20

Description: Part of proof of Lemma K of Crawley p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be f_i. Our D , C , O , Q , U , V represent their f_1, f_2, k_1, k_2, sigma_1, sigma_2. (Contributed by NM, 5-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	$⊢ U = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
	cdlemk2a.q	$⊢ Q = S (C)$
Assertion	cdlemk20	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to U (C) (P) = Q (P)$

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		cdlemk2a.q	$⊢ Q = S (C)$
13		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to (K \in HL \land W \in H)$
14		simp23	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to R (F) = R (N)$
15		simp21r	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to C \in T$
16		simp12	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to F \in T$
17		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to D \in T$
18		simp21l	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to N \in T$
19		simp3r1	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to R (D) \neq R (F)$
20		simp3r3	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to R (C) \neq R (D)$
21	20	necomd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to R (D) \neq R (C)$
22	19 21	jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to (R (D) \neq R (F) \land R (D) \neq R (C))$
23		simp3l1	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to F \neq {I ↾}_{B}$
24		simp3l3	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to C \neq {I ↾}_{B}$
25		simp3l2	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to D \neq {I ↾}_{B}$
26	23 24 25	3jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land D \neq {I ↾}_{B})$
27		simp22	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
28	1 2 3 4 5 6 7 8 9 10 11	cdlemkuv2	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land C \in T) \land (F \in T \land D \in T \land N \in T) \land ((R (D) \neq R (F) \land R (D) \neq R (C)) \land (F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to U (C) (P) = (P \underset{˙}{\lor} R (C)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (C \circ D^{-1}))$
29	13 14 15 16 17 18 22 26 27 28	syl333anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to U (C) (P) = (P \underset{˙}{\lor} R (C)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (C \circ D^{-1}))$
30	2 3 5 6 7 8	trljat1	$⊢ ((K \in HL \land W \in H) \land C \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to P \underset{˙}{\lor} R (C) = P \underset{˙}{\lor} C (P)$
31	13 15 27 30	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to P \underset{˙}{\lor} R (C) = P \underset{˙}{\lor} C (P)$
32	10	fveq1i	$⊢ O (P) = S (D) (P)$
33	32	a1i	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to O (P) = S (D) (P)$
34	6 7 8	trlcocnv	$⊢ ((K \in HL \land W \in H) \land C \in T \land D \in T) \to R (C \circ D^{-1}) = R (D \circ C^{-1})$
35	13 15 17 34	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to R (C \circ D^{-1}) = R (D \circ C^{-1})$
36	33 35	oveq12d	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to O (P) \underset{˙}{\lor} R (C \circ D^{-1}) = S (D) (P) \underset{˙}{\lor} R (D \circ C^{-1})$
37	31 36	oveq12d	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to (P \underset{˙}{\lor} R (C)) \underset{˙}{\land} (O (P) \underset{˙}{\lor} R (C \circ D^{-1})) = (P \underset{˙}{\lor} C (P)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (D \circ C^{-1}))$
38	12	fveq1i	$⊢ Q (P) = S (C) (P)$
39	18 17	jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to (N \in T \land D \in T)$
40		simp3r2	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to R (C) \neq R (F)$
41	40 19	jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to (R (C) \neq R (F) \land R (D) \neq R (F))$
42	1 2 3 5 6 7 8 4 9	cdlemk12	$⊢ (((K \in HL \land W \in H) \land F \in T \land C \in T) \land ((N \in T \land D \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land C \neq {I ↾}_{B} \land D \neq {I ↾}_{B}) \land (R (C) \neq R (F) \land R (D) \neq R (F)) \land R (C) \neq R (D))) \to S (C) (P) = (P \underset{˙}{\lor} C (P)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (D \circ C^{-1}))$
43	13 16 15 39 27 14 26 41 20 42	syl333anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to S (C) (P) = (P \underset{˙}{\lor} C (P)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (D \circ C^{-1}))$
44	38 43	eqtr2id	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to (P \underset{˙}{\lor} C (P)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (D \circ C^{-1})) = Q (P)$
45	29 37 44	3eqtrd	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land C \neq {I ↾}_{B}) \land (R (D) \neq R (F) \land R (C) \neq R (F) \land R (C) \neq R (D)))) \to U (C) (P) = Q (P)$

Metamath Proof Explorer

Theorem cdlemk20

Proof