cdlemk54

Description: Part of proof of Lemma K of Crawley p. 118. Line 10, p. 120. G , I stand for g, h. X represents tau. (Contributed by NM, 26-Jul-2013)

	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	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
Assertion	cdlemk54	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X = (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X$

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		coass	$⊢ (G \circ I) \circ j = G \circ (I \circ j)$
13		csbeq1	$⊢ (G \circ I) \circ j = G \circ (I \circ j) \to ⦋ ((G \circ I) \circ j) / g ⦌ X = ⦋ (G \circ (I \circ j)) / g ⦌ X$
14	12 13	ax-mp	$⊢ ⦋ ((G \circ I) \circ j) / g ⦌ X = ⦋ (G \circ (I \circ j)) / g ⦌ X$
15		simp1	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
16		simp21	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (F \in T \land F \neq {I ↾}_{B} \land N \in T)$
17		simp1l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (K \in HL \land W \in H)$
18		simp22	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to G \in T$
19		simp31l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to I \in T$
20	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land G \in T \land I \in T) \to G \circ I \in T$
21	17 18 19 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to G \circ I \in T$
22		simp23	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
23		simp32	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to j \in T$
24		simp333	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (j) \neq R (G \circ I)$
25	24	necomd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (G \circ I) \neq R (j)$
26	1 2 3 4 5 6 7 8 9 10 11	cdlemk53	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \circ I \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (j \in T \land R (G \circ I) \neq R (j))) \to ⦋ ((G \circ I) \circ j) / g ⦌ X = ⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X$
27	15 16 21 22 23 25 26	syl132anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ ((G \circ I) \circ j) / g ⦌ X = ⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X$
28		simp2	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W))$
29	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land I \in T \land j \in T) \to I \circ j \in T$
30	17 19 23 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to I \circ j \in T$
31		simp31r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (G) = R (I)$
32		simp332	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (j) \neq R (G)$
33	32 31	neeqtrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (j) \neq R (I)$
34	33	necomd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (I) \neq R (j)$
35		simp331	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to j \neq {I ↾}_{B}$
36	1 6 7 8	trlcone	$⊢ ((K \in HL \land W \in H) \land (I \in T \land j \in T) \land (R (I) \neq R (j) \land j \neq {I ↾}_{B})) \to R (I) \neq R (I \circ j)$
37	17 19 23 34 35 36	syl122anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (I) \neq R (I \circ j)$
38	31 37	eqnetrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (G) \neq R (I \circ j)$
39	1 2 3 4 5 6 7 8 9 10 11	cdlemk53	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \circ j \in T \land R (G) \neq R (I \circ j))) \to ⦋ (G \circ (I \circ j)) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ (I \circ j) / g ⦌ X$
40	15 28 30 38 39	syl112anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ (I \circ j)) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ (I \circ j) / g ⦌ X$
41	1 2 3 4 5 6 7 8 9 10 11	cdlemk53	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land I \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (j \in T \land R (I) \neq R (j))) \to ⦋ (I \circ j) / g ⦌ X = ⦋ I / g ⦌ X \circ ⦋ j / g ⦌ X$
42	15 16 19 22 23 34 41	syl132anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (I \circ j) / g ⦌ X = ⦋ I / g ⦌ X \circ ⦋ j / g ⦌ X$
43	42	coeq2d	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ G / g ⦌ X \circ ⦋ (I \circ j) / g ⦌ X = ⦋ G / g ⦌ X \circ (⦋ I / g ⦌ X \circ ⦋ j / g ⦌ X)$
44		coass	$⊢ (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X = ⦋ G / g ⦌ X \circ (⦋ I / g ⦌ X \circ ⦋ j / g ⦌ X)$
45	43 44	eqtr4di	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ G / g ⦌ X \circ ⦋ (I \circ j) / g ⦌ X = (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X$
46	40 45	eqtrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ (I \circ j)) / g ⦌ X = (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X$
47	14 27 46	3eqtr3a	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X = (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X$

Metamath Proof Explorer

Theorem cdlemk54

Proof