cdlemk19x

Description: cdlemk19 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-Jul-2013)

	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	cdlemk19x	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to ⦋ F / g ⦌ X (P) = N (P)$

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	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (K \in HL \land W \in H)$
13	1 6 7 8	cdlemftr1	$⊢ (K \in HL \land W \in H) \to \exists b \in T (b \neq {I ↾}_{B} \land R (b) \neq R (F))$
14	12 13	syl	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to \exists b \in T (b \neq {I ↾}_{B} \land R (b) \neq R (F))$
15		nfv	$⊢ Ⅎ b (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W))$
16		nfcv	$⊢ \underline{Ⅎ} b F$
17		nfra1	$⊢ Ⅎ b \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y)$
18		nfcv	$⊢ \underline{Ⅎ} b T$
19	17 18	nfriota	$⊢ \underline{Ⅎ} b (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (g)) \to z (P) = Y))$
20	11 19	nfcxfr	$⊢ \underline{Ⅎ} b X$
21	16 20	nfcsbw	$⊢ \underline{Ⅎ} b ⦋ F / g ⦌ X$
22		nfcv	$⊢ \underline{Ⅎ} b P$
23	21 22	nffv	$⊢ \underline{Ⅎ} b ⦋ F / g ⦌ X (P)$
24	23	nfeq1	$⊢ Ⅎ b ⦋ F / g ⦌ X (P) = N (P)$
25		simpl1	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ((K \in HL \land W \in H) \land R (F) = R (N))$
26		simpl2	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (F \in T \land F \neq {I ↾}_{B} \land N \in T)$
27		simpl3	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
28		simpr	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))$
29	1 2 3 4 5 6 7 8 9 10 11	cdlemk19xlem	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ X (P) = N (P)$
30	25 26 27 28 29	syl121anc	$⊢ ((((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F)))) \to ⦋ F / g ⦌ X (P) = N (P)$
31	30	exp32	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (b \in T \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F)) \to ⦋ F / g ⦌ X (P) = N (P)))$
32	15 24 31	rexlimd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to (\exists b \in T (b \neq {I ↾}_{B} \land R (b) \neq R (F)) \to ⦋ F / g ⦌ X (P) = N (P))$
33	14 32	mpd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land (F \in T \land F \neq {I ↾}_{B} \land N \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to ⦋ F / g ⦌ X (P) = N (P)$

Metamath Proof Explorer

Theorem cdlemk19x

Proof