cdlemk22-3

Description: Part of proof of Lemma K of Crawley p. 118. Lines 26-27, p. 119 for i=1 and j=2. (Contributed by NM, 7-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	cdlemk22-3	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to (D Y G) (P) = (C Y G) (P)$

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		eqid	$⊢ S (C) = S (C)$
12		eqid	$⊢ (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (C) (P) \underset{˙}{\lor} R (e \circ C^{-1})))) = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (C) (P) \underset{˙}{\lor} R (e \circ C^{-1}))))$
13		eqid	$⊢ S (D) = S (D)$
14		eqid	$⊢ (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1}))))$
15	1 2 3 4 5 6 7 8 9 11 12 13 14	cdlemk22	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) (G) (P) = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (C) (P) \underset{˙}{\lor} R (e \circ C^{-1})))) (G) (P)$
16		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to D \in T$
17		simp212	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to G \in T$
18	1 2 3 4 5 6 7 8 9 10 13 14	cdlemkuu	$⊢ (D \in T \land G \in T) \to D Y G = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) (G)$
19	16 17 18	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to D Y G = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) (G)$
20	19	fveq1d	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to (D Y G) (P) = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (D) (P) \underset{˙}{\lor} R (e \circ D^{-1})))) (G) (P)$
21		simp213	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to C \in T$
22	1 2 3 4 5 6 7 8 9 10 11 12	cdlemkuu	$⊢ (C \in T \land G \in T) \to C Y G = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (C) (P) \underset{˙}{\lor} R (e \circ C^{-1})))) (G)$
23	21 17 22	syl2anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to C Y G = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (C) (P) \underset{˙}{\lor} R (e \circ C^{-1})))) (G)$
24	23	fveq1d	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to (C Y G) (P) = (e \in T ⟼ (ι j \in T \| j (P) = (P \underset{˙}{\lor} R (e)) \underset{˙}{\land} (S (C) (P) \underset{˙}{\lor} R (e \circ C^{-1})))) (G) (P)$
25	15 20 24	3eqtr4d	$⊢ (((K \in HL \land W \in H) \land F \in T \land D \in T) \land ((N \in T \land G \in T \land C \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land ((F \neq {I ↾}_{B} \land D \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (C \neq {I ↾}_{B} \land R (G) \neq R (C) \land R (C) \neq R (F)) \land (R (D) \neq R (F) \land R (G) \neq R (D) \land R (C) \neq R (D)))) \to (D Y G) (P) = (C Y G) (P)$

Metamath Proof Explorer

Theorem cdlemk22-3

Proof