cdlemk55b

	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	cdlemk55b	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

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		simp1ll	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \to K \in HL$
13		simp1lr	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \to W \in H$
14	1 6 7 8	cdlemftr2	$⊢ (K \in HL \land W \in H) \to \exists j \in T (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))$
15	12 13 14	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \to \exists j \in T (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))$
16		simp11	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
17		simp12	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))) \to ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W))$
18		simp13	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))) \to (I \in T \land R (G) = R (I))$
19		simp2	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))) \to j \in T$
20		simp3	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))) \to (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))$
21	1 2 3 4 5 6 7 8 9 10 11	cdlemk55a	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land ((I \in T \land R (G) = R (I)) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
22	16 17 18 19 20 21	syl113anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \land j \in T \land (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
23	22	rexlimdv3a	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \to (\exists j \in T (j \neq {I ↾}_{B} \land R (j) \neq R (G) \land R (j) \neq R (G \circ I)) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X)$
24	15 23	mpd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (I \in T \land R (G) = R (I))) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$

Metamath Proof Explorer

Theorem cdlemk55b

Proof