cdlemk19ylem

	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	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
	cdlemk5c.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk5a.u2	$⊢ C = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1}))))$
Assertion	cdlemk19ylem	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ Y = N (P)$

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		cdlemk5c.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
12		cdlemk5a.u2	$⊢ C = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1}))))$
13		simp1l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (K \in HL \land W \in H)$
14		simp1r	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (F \in T \land F \neq {I ↾}_{B})$
15		simp2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))$
16		simp3l	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to b \in T$
17		simp3rl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to b \neq {I ↾}_{B}$
18		simp3rr	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to R (b) \neq R (F)$
19	17 18 18	3jca	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (F))$
20	1 2 3 4 5 6 7 8 9 10 11 12	cdlemkyuu	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ Y = C (F) (P)$
21	13 14 14 15 16 19 20	syl312anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ Y = C (F) (P)$
22		simp1rl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to F \in T$
23		simp1rr	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to F \neq {I ↾}_{B}$
24		eqid	$⊢ S (b) = S (b)$
25	1 2 3 4 5 6 7 8 11 24 12	cdlemk19	$⊢ (((K \in HL \land W \in H) \land F \in T \land b \in T) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land R (b) \neq R (F))) \to C (F) = N$
26	13 22 16 15 23 17 18 25	syl313anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to C (F) = N$
27	26	fveq1d	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to C (F) (P) = N (P)$
28	21 27	eqtrd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ Y = N (P)$

Metamath Proof Explorer

Theorem cdlemk19ylem

Proof