cdlemk43N

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. (Contributed by NM, 31-Jul-2013) (New usage is discouraged.)

	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	cdlemk43N	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to U (G) (P) = ⦋ G / g ⦌ Y$

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		simp213	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \neq N$
14		simp22l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to G \in T$
15	11 12	cdlemk40f	$⊢ (F \neq N \land G \in T) \to U (G) = ⦋ G / g ⦌ X$
16	13 14 15	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to U (G) = ⦋ G / g ⦌ X$
17	16	fveq1d	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to U (G) (P) = ⦋ G / g ⦌ X (P)$
18		simp1l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (K \in HL \land W \in H)$
19		simp211	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \in T$
20		simp212	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to N \in T$
21		simp1r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to R (F) = R (N)$
22	1 6 7 8	trlnid	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (F \neq N \land R (F) = R (N))) \to F \neq {I ↾}_{B}$
23	18 19 20 13 21 22	syl122anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to F \neq {I ↾}_{B}$
24	19 23	jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (F \in T \land F \neq {I ↾}_{B})$
25		simp22	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (G \in T \land G \neq {I ↾}_{B})$
26		simp23	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
27		simp3	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))$
28	1 2 3 4 5 6 7 8 9 10 11	cdlemk42	$⊢ (((K \in HL \land W \in H) \land (F \in T \land F \neq {I ↾}_{B}) \land (G \in T \land G \neq {I ↾}_{B})) \land (N \in T \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y$
29	18 24 25 20 26 21 27 28	syl331anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to ⦋ G / g ⦌ X (P) = ⦋ G / g ⦌ Y$
30	17 29	eqtrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land N \in T \land F \neq N) \land (G \in T \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \land (b \in T \land (b \neq {I ↾}_{B} \land R (b) \neq R (F) \land R (b) \neq R (G)))) \to U (G) (P) = ⦋ G / g ⦌ Y$

Metamath Proof Explorer

Theorem cdlemk43N

Proof