cdlemk53

Description: Part of proof of Lemma K of Crawley p. 118. Line 7, 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	cdlemk53	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

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		simp1l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to (K \in HL \land W \in H)$
13		simp211	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to F \in T$
14		simp212	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to F \neq {I ↾}_{B}$
15	13 14	jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to (F \in T \land F \neq {I ↾}_{B})$
16		simp22	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to G \in T$
17		simp213	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to N \in T$
18		simp23	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
19		simp1r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to R (F) = R (N)$
20	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s-id	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ G / g ⦌ X \in T$
21	12 15 16 17 18 19 20	syl132anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X \in T$
22	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T) \to ⦋ G / g ⦌ X : B \underset{1-1 onto}{⟶} B$
23	12 21 22	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X : B \underset{1-1 onto}{⟶} B$
24		f1of	$⊢ ⦋ G / g ⦌ X : B \underset{1-1 onto}{⟶} B \to ⦋ G / g ⦌ X : B ⟶ B$
25		fcoi1	$⊢ ⦋ G / g ⦌ X : B ⟶ B \to ⦋ G / g ⦌ X \circ ({I ↾}_{B}) = ⦋ G / g ⦌ X$
26	23 24 25	3syl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to ⦋ G / g ⦌ X \circ ({I ↾}_{B}) = ⦋ G / g ⦌ X$
27	26	adantr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to ⦋ G / g ⦌ X \circ ({I ↾}_{B}) = ⦋ G / g ⦌ X$
28		simpl1l	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to (K \in HL \land W \in H)$
29	13 17 19	3jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to (F \in T \land N \in T \land R (F) = R (N))$
30	29	adantr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to (F \in T \land N \in T \land R (F) = R (N))$
31		simpl23	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
32		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to I = {I ↾}_{B}$
33	1 2 3 4 5 6 7 8 9 10 11	cdlemkid	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T \land R (F) = R (N)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land I = {I ↾}_{B})) \to ⦋ I / g ⦌ X = {I ↾}_{B}$
34	28 30 31 32 33	syl112anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to ⦋ I / g ⦌ X = {I ↾}_{B}$
35	34	coeq2d	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X = ⦋ G / g ⦌ X \circ ({I ↾}_{B})$
36	32	coeq2d	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to G \circ I = G \circ ({I ↾}_{B})$
37	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land G \in T) \to G : B \underset{1-1 onto}{⟶} B$
38	12 16 37	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to G : B \underset{1-1 onto}{⟶} B$
39		f1of	$⊢ G : B \underset{1-1 onto}{⟶} B \to G : B ⟶ B$
40		fcoi1	$⊢ G : B ⟶ B \to G \circ ({I ↾}_{B}) = G$
41	38 39 40	3syl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to G \circ ({I ↾}_{B}) = G$
42	41	adantr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to G \circ ({I ↾}_{B}) = G$
43	36 42	eqtrd	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to G \circ I = G$
44	43	csbeq1d	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X$
45	27 35 44	3eqtr4rd	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I = {I ↾}_{B}) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
46		simpl1	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I \neq {I ↾}_{B}) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
47		simpl2	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I \neq {I ↾}_{B}) \to ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W))$
48		simpl3l	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I \neq {I ↾}_{B}) \to I \in T$
49		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I \neq {I ↾}_{B}) \to I \neq {I ↾}_{B}$
50		simpl3r	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I \neq {I ↾}_{B}) \to R (G) \neq R (I)$
51	1 2 3 4 5 6 7 8 9 10 11	cdlemk53b	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
52	46 47 48 49 50 51	syl113anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \land I \neq {I ↾}_{B}) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
53	45 52	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

Metamath Proof Explorer

Theorem cdlemk53

Proof