cdlemk55a

	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	cdlemk55a	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

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	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (K \in HL \land W \in H)$
13		simp211	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to F \in T$
14		simp212	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to F \neq {I ↾}_{B}$
15	13 14	jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (F \in T \land F \neq {I ↾}_{B})$
16		simp32	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to j \in T$
17		simp213	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to N \in T$
18		simp23	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
19		simp1r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to R (F) = R (N)$
20	18 19	jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))$
21	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s-id	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land j \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ j / g ⦌ X \in T$
22	12 15 16 17 20 21	syl131anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ j / g ⦌ X \in T$
23	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land ⦋ j / g ⦌ X \in T) \to ⦋ j / g ⦌ X : B \underset{1-1 onto}{⟶} B$
24	12 22 23	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ j / g ⦌ X : B \underset{1-1 onto}{⟶} B$
25		f1ococnv2	$⊢ ⦋ j / g ⦌ X : B \underset{1-1 onto}{⟶} B \to ⦋ j / g ⦌ X \circ {⦋ j / g ⦌ X}^{-1} = {I ↾}_{B}$
26	24 25	syl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ j / g ⦌ X \circ {⦋ j / g ⦌ X}^{-1} = {I ↾}_{B}$
27	26	coeq2d	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X \circ (⦋ j / g ⦌ X \circ {⦋ j / g ⦌ X}^{-1}) = ⦋ (G \circ I) / g ⦌ X \circ ({I ↾}_{B})$
28		simp22	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to G \in T$
29		simp31l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to I \in T$
30	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land G \in T \land I \in T) \to G \circ I \in T$
31	12 28 29 30	syl3anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to G \circ I \in T$
32	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s-id	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \circ I \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ (G \circ I) / g ⦌ X \in T$
33	12 15 31 17 20 32	syl131anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X \in T$
34	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land ⦋ (G \circ I) / g ⦌ X \in T) \to ⦋ (G \circ I) / g ⦌ X : B \underset{1-1 onto}{⟶} B$
35	12 33 34	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X : B \underset{1-1 onto}{⟶} B$
36		f1of	$⊢ ⦋ (G \circ I) / g ⦌ X : B \underset{1-1 onto}{⟶} B \to ⦋ (G \circ I) / g ⦌ X : B ⟶ B$
37		fcoi1	$⊢ ⦋ (G \circ I) / g ⦌ X : B ⟶ B \to ⦋ (G \circ I) / g ⦌ X \circ ({I ↾}_{B}) = ⦋ (G \circ I) / g ⦌ X$
38	35 36 37	3syl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X \circ ({I ↾}_{B}) = ⦋ (G \circ I) / g ⦌ X$
39	27 38	eqtr2d	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ (G \circ I) / g ⦌ X \circ (⦋ j / g ⦌ X \circ {⦋ j / g ⦌ X}^{-1})$
40		coass	$⊢ (⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X) \circ {⦋ j / g ⦌ X}^{-1} = ⦋ (G \circ I) / g ⦌ X \circ (⦋ j / g ⦌ X \circ {⦋ j / g ⦌ X}^{-1})$
41	39 40	eqtr4di	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X = (⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X) \circ {⦋ j / g ⦌ X}^{-1}$
42	1 2 3 4 5 6 7 8 9 10 11	cdlemk54	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X = (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X$
43	42	coeq1d	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X) \circ {⦋ j / g ⦌ X}^{-1} = ((⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X) \circ {⦋ j / g ⦌ X}^{-1}$
44		coass	$⊢ ((⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X) \circ {⦋ j / g ⦌ X}^{-1} = (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ (⦋ j / g ⦌ X \circ {⦋ j / g ⦌ X}^{-1})$
45	26	coeq2d	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ (⦋ j / g ⦌ X \circ {⦋ j / g ⦌ X}^{-1}) = (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ({I ↾}_{B})$
46	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s-id	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ G / g ⦌ X \in T$
47	12 15 28 17 20 46	syl131anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ G / g ⦌ X \in T$
48	1 2 3 4 5 6 7 8 9 10 11	cdlemk35s-id	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land I \in T \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ⦋ I / g ⦌ X \in T$
49	12 15 29 17 20 48	syl131anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ I / g ⦌ X \in T$
50	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \in T \land ⦋ I / g ⦌ X \in T) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X \in T$
51	12 47 49 50	syl3anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X \in T$
52	1 6 7	ltrn1o	$⊢ ((K \in HL \land W \in H) \land ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X \in T) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) : B \underset{1-1 onto}{⟶} B$
53	12 51 52	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) : B \underset{1-1 onto}{⟶} B$
54		f1of	$⊢ (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) : B \underset{1-1 onto}{⟶} B \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) : B ⟶ B$
55		fcoi1	$⊢ (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) : B ⟶ B \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ({I ↾}_{B}) = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
56	53 54 55	3syl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ({I ↾}_{B}) = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
57	45 56	eqtrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ (⦋ j / g ⦌ X \circ {⦋ j / g ⦌ X}^{-1}) = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
58	44 57	syl5eq	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ((⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X) \circ ⦋ j / g ⦌ X) \circ {⦋ j / g ⦌ X}^{-1} = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
59	43 58	eqtrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to (⦋ (G \circ I) / g ⦌ X \circ ⦋ j / g ⦌ X) \circ {⦋ j / g ⦌ X}^{-1} = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
60	41 59	eqtrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

Metamath Proof Explorer

Theorem cdlemk55a

Proof