cdlemk56

Description: Part of Lemma K of Crawley p. 118. Line 11, p. 120, "tau is in Delta" i.e. U is a trace-preserving endormorphism. (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	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
	cdlemk5.e	$⊢ E = TEndo (K) (W)$
Assertion	cdlemk56	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U \in E$

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		cdlemk5.e	$⊢ E = TEndo (K) (W)$
14		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
15		vex	$⊢ g \in V$
16		riotaex	$⊢ (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y)) \in V$
17	11 16	eqeltri	$⊢ X \in V$
18	15 17	ifex	$⊢ if (F = N, g, X) \in V$
19	18	rgenw	$⊢ \forall g \in T if (F = N, g, X) \in V$
20	12	fnmpt	$⊢ \forall g \in T if (F = N, g, X) \in V \to U Fn T$
21	19 20	mp1i	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U Fn T$
22		simpl11	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to (K \in HL \land W \in H)$
23		simpl2	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to R (F) = R (N)$
24		simpl12	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to F \in T$
25		simpl13	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to N \in T$
26		simpr	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to f \in T$
27		simpl3	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
28	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk35u	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land N \in T \land f \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (f) \in T$
29	22 23 24 25 26 27 28	syl231anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to U (f) \in T$
30	29	ralrimiva	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \forall f \in T U (f) \in T$
31		ffnfv	$⊢ U : T ⟶ T \leftrightarrow (U Fn T \land \forall f \in T U (f) \in T)$
32	21 30 31	sylanbrc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U : T ⟶ T$
33		simp11	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T \land h \in T) \to ((K \in HL \land W \in H) \land F \in T \land N \in T)$
34		simp12	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T \land h \in T) \to R (F) = R (N)$
35		simp2	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T \land h \in T) \to f \in T$
36		simp3	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T \land h \in T) \to h \in T$
37		simp13	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T \land h \in T) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
38	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk55u	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land f \in T \land h \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (f \circ h) = U (f) \circ U (h)$
39	33 34 35 36 37 38	syl131anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T \land h \in T) \to U (f \circ h) = U (f) \circ U (h)$
40		simpl1	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to ((K \in HL \land W \in H) \land F \in T \land N \in T)$
41	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk39u	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land (R (F) = R (N) \land f \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (U (f)) \underset{˙}{\leq} R (f)$
42	40 23 26 27 41	syl121anc	$⊢ ((((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land f \in T) \to R (U (f)) \underset{˙}{\leq} R (f)$
43	2 6 7 8 13 14 32 39 42	istendod	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U \in E$

Metamath Proof Explorer

Theorem cdlemk56

Proof