cdlemk36

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. (Contributed by NM, 18-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk4.b	$⊢ B = {Base}_{K}$
	cdlemk4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk4.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk4.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk4.a	$⊢ A = Atoms (K)$
	cdlemk4.h	$⊢ H = LHyp (K)$
	cdlemk4.t	$⊢ T = LTrn (K) (W)$
	cdlemk4.r	$⊢ R = trL (K) (W)$
	cdlemk4.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
	cdlemk4.y	$⊢ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
	cdlemk4.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	cdlemk36	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to X (P) = Y$

Proof

Step	Hyp	Ref	Expression
1		cdlemk4.b	$⊢ B = {Base}_{K}$
2		cdlemk4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk4.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk4.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk4.a	$⊢ A = Atoms (K)$
6		cdlemk4.h	$⊢ H = LHyp (K)$
7		cdlemk4.t	$⊢ T = LTrn (K) (W)$
8		cdlemk4.r	$⊢ R = trL (K) (W)$
9		cdlemk4.z	$⊢ Z = (P \underset{˙}{\lor} R (b)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (b \circ F^{-1}))$
10		cdlemk4.y	$⊢ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
11		cdlemk4.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	11	eqcomi	$⊢ (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)) = X$
13		simpl1	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (K \in HL \land W \in H)$
14		simpl2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (F \in T \land F \neq {I ↾}_{B})$
15		simpl3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (G \in T \land G \neq {I ↾}_{B})$
16		simpr1	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to N \in T$
17		simpr2	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18		simpr3	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to R (F) = R (N)$
19	1 2 3 4 5 6 7 8 9 10 11	cdlemk35	$⊢ ((K \in HL \land W \in H) \land ((F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B}) \land N \in T) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X \in T$
20	13 14 15 16 17 18 19	syl132anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to X \in T$
21	11 20	eqeltrrid	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)) \in T$
22	7	fvexi	$⊢ T \in V$
23	22	riotaclbBAD	$⊢ \exists! z \in T \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y) \leftrightarrow (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)) \in T$
24	21 23	sylibr	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to \exists! z \in T \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)$
25		nfriota1	$⊢ \underline{Ⅎ} z (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y))$
26	11 25	nfcxfr	$⊢ \underline{Ⅎ} z X$
27		nfcv	$⊢ \underline{Ⅎ} z T$
28		nfv	$⊢ Ⅎ z (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G))$
29		nfcv	$⊢ \underline{Ⅎ} z P$
30	26 29	nffv	$⊢ \underline{Ⅎ} z X (P)$
31	30	nfeq1	$⊢ Ⅎ z X (P) = Y$
32	28 31	nfim	$⊢ Ⅎ z ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y)$
33	27 32	nfralw	$⊢ Ⅎ z \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y)$
34		nfra1	$⊢ Ⅎ b \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)$
35		nfcv	$⊢ \underline{Ⅎ} b T$
36	34 35	nfriota	$⊢ \underline{Ⅎ} b (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y))$
37	11 36	nfcxfr	$⊢ \underline{Ⅎ} b X$
38	37	nfeq2	$⊢ Ⅎ b z = X$
39		fveq1	$⊢ z = X \to z (P) = X (P)$
40	39	eqeq1d	$⊢ z = X \to (z (P) = Y \leftrightarrow X (P) = Y)$
41	40	imbi2d	$⊢ z = X \to (((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y) \leftrightarrow ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y))$
42	38 41	ralbid	$⊢ z = X \to (\forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y) \leftrightarrow \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y))$
43	26 33 42	riota2f	$⊢ (X \in T \land \exists! z \in T \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)) \to (\forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y) \leftrightarrow (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)) = X)$
44	20 24 43	syl2anc	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (\forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y) \leftrightarrow (ι z \in T \| \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to z (P) = Y)) = X)$
45	12 44	mpbiri	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y)$
46		rsp	$⊢ \forall b \in T ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y) \to (b \in T \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y))$
47	45 46	syl	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to (b \in T \to ((b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)) \to X (P) = Y))$
48	47	impd	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N))) \to ((b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G))) \to X (P) = Y)$
49	48	3impia	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to X (P) = Y$

Metamath Proof Explorer

Theorem cdlemk36

Proof