cdlemk19xlem

	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	cdlemk19xlem	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ X (P) = N (P)$

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		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	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (K \in HL \land W \in H)$
13		simp2l1	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to F \in T$
14		simp2l2	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to F \neq {I ↾}_{B}$
15	13 14	jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (F \in T \land F \neq {I ↾}_{B})$
16		simp2l3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to N \in T$
17		simp2r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18		simp1r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to R (F) = R (N)$
19		simp3l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to b \in T$
20		simp3rl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to b \neq {I ↾}_{B}$
21		simp3rr	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to R (b) \neq R (F)$
22	20 21 21	3jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (F))$
23	1 2 3 4 5 6 7 8 9 10 11	cdlemk42	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ X (P) = ⦋ F / g ⦌ Y$
24	12 15 15 16 17 18 19 22 23	syl332anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ X (P) = ⦋ F / g ⦌ Y$
25		simp3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))$
26	1 2 3 4 5 6 7 8 9 10	cdlemk19y	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ Y = N (P)$
27	12 15 16 17 18 25 26	syl231anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ Y = N (P)$
28	24 27	eqtrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ X (P) = N (P)$

Metamath Proof Explorer

Theorem cdlemk19xlem

Proof