cdlemk53b

	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	cdlemk53b	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

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	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (K \in HL \land W \in H)$
13		simp211	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to F \in T$
14		simp212	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to F \neq {I ↾}_{B}$
15	13 14	jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (F \in T \land F \neq {I ↾}_{B})$
16		simp31	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to I \in T$
17		simp213	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to N \in T$
18		simp23	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
19		simp1r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to R (F) = R (N)$
20	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s-id	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land I \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ I / g ⦌ X \in T$
21	12 15 16 17 18 19 20	syl132anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X \in T$
22	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land ⦋ I / g ⦌ X \in T) \to ⦋ I / g ⦌ X : B \underset{1-1 onto}{⟶} B$
23	12 21 22	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ I / g ⦌ X : B \underset{1-1 onto}{⟶} B$
24	23	adantr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to ⦋ I / g ⦌ X : B \underset{1-1 onto}{⟶} B$
25		f1of	$⊢ ⦋ I / g ⦌ X : B \underset{1-1 onto}{⟶} B \to ⦋ I / g ⦌ X : B ⟶ B$
26		fcoi2	$⊢ ⦋ I / g ⦌ X : B ⟶ B \to ({I ↾}_{B}) \circ ⦋ I / g ⦌ X = ⦋ I / g ⦌ X$
27	24 25 26	3syl	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to ({I ↾}_{B}) \circ ⦋ I / g ⦌ X = ⦋ I / g ⦌ X$
28		simpl1l	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to (K \in HL \land W \in H)$
29	13 17 19	3jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to (F \in T \land N \in T \land R (F) = R (N))$
30	29	adantr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to (F \in T \land N \in T \land R (F) = R (N))$
31		simpl23	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
32		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to G = {I ↾}_{B}$
33	1 2 3 4 5 6 7 8 9 10 11	cdlemkid	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T \land R (F) = R (N)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land G = {I ↾}_{B})) \to ⦋ G / g ⦌ X = {I ↾}_{B}$
34	28 30 31 32 33	syl112anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to ⦋ G / g ⦌ X = {I ↾}_{B}$
35	34	coeq1d	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X = ({I ↾}_{B}) \circ ⦋ I / g ⦌ X$
36	32	coeq1d	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to G \circ I = ({I ↾}_{B}) \circ I$
37		simpl31	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to I \in T$
38	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land I \in T) \to I : B \underset{1-1 onto}{⟶} B$
39	28 37 38	syl2anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to I : B \underset{1-1 onto}{⟶} B$
40		f1of	$⊢ I : B \underset{1-1 onto}{⟶} B \to I : B ⟶ B$
41		fcoi2	$⊢ I : B ⟶ B \to ({I ↾}_{B}) \circ I = I$
42	39 40 41	3syl	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to ({I ↾}_{B}) \circ I = I$
43	36 42	eqtrd	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to G \circ I = I$
44	43	csbeq1d	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to ⦋ (G \circ I) / g ⦌ X = ⦋ I / g ⦌ X$
45	27 35 44	3eqtr4rd	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G = {I ↾}_{B}) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
46		simpl1l	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to (K \in HL \land W \in H)$
47	15	adantr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to (F \in T \land F \neq {I ↾}_{B})$
48		simpl22	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to G \in T$
49		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to G \neq {I ↾}_{B}$
50	48 49	jca	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to (G \in T \land G \neq {I ↾}_{B})$
51	17	adantr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to N \in T$
52		simpl23	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
53		simpl1r	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to R (F) = R (N)$
54		simpl3	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))$
55	1 2 3 4 5 6 7 8 9 10 11	cdlemk53a	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
56	46 47 50 51 52 53 54 55	syl331anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \land G \neq {I ↾}_{B}) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
57	45 56	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land I \neq {I ↾}_{B} \land R (G) \neq R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

Metamath Proof Explorer

Theorem cdlemk53b

Proof