cdlemk55u1

	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	$⊢ U = (g \in T ⟼ if (F = N, g, X))$
Assertion	cdlemk55u1	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G \circ I) = U (G) \circ U (I)$

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		simp11	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
14		simp21l	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to R (F) = R (N)$
15		simp12	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to F \in T$
16		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to N \in T$
17		simp21r	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to F \neq N$
18	1 6 7 8	trlnid	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (F \neq N \land R (F) = R (N))) \to F \neq {I ↾}_{B}$
19	13 15 16 17 14 18	syl122anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to F \neq {I ↾}_{B}$
20	15 19 16	3jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (F \in T \land F \neq {I ↾}_{B} \land N \in T)$
21		simp22	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to G \in T$
22		simp23	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to I \in T$
23		simp3	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
24	1 2 3 4 5 6 7 8 9 10 11	cdlemk55	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
25	13 14 20 21 22 23 24	syl231anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to ⦋ (G \circ I) / g ⦌ X = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
26	6 7	ltrnco	$⊢ ((K \in HL \land W \in H) \land G \in T \land I \in T) \to G \circ I \in T$
27	13 21 22 26	syl3anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to G \circ I \in T$
28	11 12	cdlemk40f	$⊢ (F \neq N \land G \circ I \in T) \to U (G \circ I) = ⦋ (G \circ I) / g ⦌ X$
29	17 27 28	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G \circ I) = ⦋ (G \circ I) / g ⦌ X$
30	11 12	cdlemk40f	$⊢ (F \neq N \land G \in T) \to U (G) = ⦋ G / g ⦌ X$
31	17 21 30	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G) = ⦋ G / g ⦌ X$
32	11 12	cdlemk40f	$⊢ (F \neq N \land I \in T) \to U (I) = ⦋ I / g ⦌ X$
33	17 22 32	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (I) = ⦋ I / g ⦌ X$
34	31 33	coeq12d	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G) \circ U (I) = ⦋ G / g ⦌ X \circ ⦋ I / g ⦌ X$
35	25 29 34	3eqtr4d	$⊢ (((K \in HL \land W \in H) \land F \in T \land N \in T) \land ((R (F) = R (N) \land F \neq N) \land G \in T \land I \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to U (G \circ I) = U (G) \circ U (I)$

Metamath Proof Explorer

Theorem cdlemk55u1

Proof