cdlemk31

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk3.b	$⊢ B = {Base}_{K}$
	cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemk3.a	$⊢ A = Atoms (K)$
	cdlemk3.h	$⊢ H = LHyp (K)$
	cdlemk3.t	$⊢ T = LTrn (K) (W)$
	cdlemk3.r	$⊢ R = trL (K) (W)$
	cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
	cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
Assertion	cdlemk31	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (b Y G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (G \circ b^{-1}))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk3.b	$⊢ B = {Base}_{K}$
2		cdlemk3.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk3.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk3.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemk3.a	$⊢ A = Atoms (K)$
6		cdlemk3.h	$⊢ H = LHyp (K)$
7		cdlemk3.t	$⊢ T = LTrn (K) (W)$
8		cdlemk3.r	$⊢ R = trL (K) (W)$
9		cdlemk3.s	$⊢ S = (f \in T ⟼ (ι i \in T \| i (P) = (P \underset{˙}{\lor} R (f)) \underset{˙}{\land} (N (P) \underset{˙}{\lor} R (f \circ F^{-1}))))$
10		cdlemk3.u1	$⊢ Y = (d \in T, e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (d) (P) \underset{˙}{\lor} R (e \circ d^{-1}))))$
11		simp2l2	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b \in T$
12		simp2r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \in T$
13		eqid	$⊢ S (b) = S (b)$
14		eqid	$⊢ (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1})))) = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1}))))$
15	1 2 3 4 5 6 7 8 9 10 13 14	cdlemkuu	$⊢ (b \in T \land G \in T) \to b Y G = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1})))) (G)$
16	11 12 15	syl2anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b Y G = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1})))) (G)$
17	16	fveq1d	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (b Y G) (P) = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1})))) (G) (P)$
18		simp1l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
19		simp1r	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (F) = R (N)$
20		simp2l	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (F \in T \land b \in T \land N \in T)$
21		simp31	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (R (b) \neq R (F) \land R (b) \neq R (G))$
22		simp321	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \neq {I ↾}_{B}$
23		simp323	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \neq {I ↾}_{B}$
24		simp322	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b \neq {I ↾}_{B}$
25	22 23 24	3jca	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B})$
26		simp33	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
27	1 2 3 4 5 6 7 8 9 13 14	cdlemkuv2	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N) \land G \in T) \land (F \in T \land b \in T \land N \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land b \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1})))) (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (G \circ b^{-1}))$
28	18 19 12 20 21 25 26 27	syl313anc	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (e \circ b^{-1})))) (G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (G \circ b^{-1}))$
29	17 28	eqtrd	$⊢ (((K \in HL \land W \in H) \land R (F) = R (N)) \land ((F \in T \land b \in T \land N \in T) \land G \in T) \land ((R (b) \neq R (F) \land R (b) \neq R (G)) \land (F \neq {I ↾}_{B} \land b \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (b Y G) (P) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (S (b) (P) \underset{˙}{\lor} R (G \circ b^{-1}))$

Metamath Proof Explorer

Theorem cdlemk31

Proof