cdlemk55u

Description: Part of proof of Lemma K of Crawley p. 118. Line 11, p. 120. G , I stand for g, h. X represents tau. (Contributed by NM, 31-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	$⊢ X = (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
	cdlemk5.u	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
Assertion	cdlemk55u	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G \circ I) = U (G) \circ U (I)$

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		cdlemk5.u	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
13		simpr	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to F = N$
14		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
15		simp22	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to G \in T$
16		simp23	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to I \in T$
17	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land G \in T \land I \in T) \to G \circ I \in T$
18	14 15 16 17	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to G \circ I \in T$
19	18	adantr	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to G \circ I \in T$
20	11 12	cdlemk40t	$⊢ (F = N \land G \circ I \in T) \to U (G \circ I) = G \circ I$
21	13 19 20	syl2anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (G \circ I) = G \circ I$
22		simpl22	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to G \in T$
23	11 12	cdlemk40t	$⊢ (F = N \land G \in T) \to U (G) = G$
24	13 22 23	syl2anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (G) = G$
25		simpl23	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to I \in T$
26	11 12	cdlemk40t	$⊢ (F = N \land I \in T) \to U (I) = I$
27	13 25 26	syl2anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (I) = I$
28	24 27	coeq12d	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (G) \circ U (I) = G \circ I$
29	21 28	eqtr4d	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (G \circ I) = U (G) \circ U (I)$
30		simpl1	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to ((K \in HL \land W \in H) \land F \in T \land N \in T)$
31		simpl21	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to R (F) = R (N)$
32		simpr	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to F \neq N$
33	31 32	jca	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to (R (F) = R (N) \land F \neq N)$
34		simpl22	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to G \in T$
35		simpl23	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to I \in T$
36		simpl3	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
37	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk55u1	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G \circ I) = U (G) \circ U (I)$
38	30 33 34 35 36 37	syl131anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to U (G \circ I) = U (G) \circ U (I)$
39	29 38	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G \circ I) = U (G) \circ U (I)$

Metamath Proof Explorer

Theorem cdlemk55u

Proof