cdlemk39u

Description: Part of proof of Lemma K of Crawley p. 118. Line 31, p. 119. Trace-preserving property of the value of tau, represented by ( UG ) . (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	cdlemk39u	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (U (G)) \underset{˙}{\leq} R (G)$

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	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to F = N$
14		simpl2r	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to G \in T$
15	11 12	cdlemk40t	$⊢ (F = N \land G \in T) \to U (G) = G$
16	13 14 15	syl2anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to U (G) = G$
17	16	fveq2d	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to R (U (G)) = R (G)$
18		simp11l	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to K \in HL$
19	18	hllatd	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to K \in Lat$
20		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
21		simp2r	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to G \in T$
22	1 6 7 8	trlcl	$⊢ ((K \in HL \land W \in H) \land G \in T) \to R (G) \in B$
23	20 21 22	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (G) \in B$
24	1 2	latref	$⊢ (K \in Lat \land R (G) \in B) \to R (G) \underset{˙}{\leq} R (G)$
25	19 23 24	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (G) \underset{˙}{\leq} R (G)$
26	25	adantr	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to R (G) \underset{˙}{\leq} R (G)$
27	17 26	eqbrtrd	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F = N) \to R (U (G)) \underset{˙}{\leq} R (G)$
28		simpl1	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to ((K \in HL \land W \in H) \land F \in T \land N \in T)$
29		simpl2l	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to R (F) = R (N)$
30		simpr	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to F \neq N$
31		simpl2r	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to G \in T$
32		simpl3	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
33	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk39u1	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land F \neq N \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (U (G)) \underset{˙}{\leq} R (G)$
34	28 29 30 31 32 33	syl131anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land F \neq N) \to R (U (G)) \underset{˙}{\leq} R (G)$
35	27 34	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (U (G)) \underset{˙}{\leq} R (G)$

Metamath Proof Explorer

Theorem cdlemk39u

Proof