cdlemk5

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 25-Jun-2013)

	Ref	Expression
Hypotheses	cdlemk.b	$⊢ B = {Base}_{K}$
	cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemk.a	$⊢ A = Atoms (K)$
	cdlemk.h	$⊢ H = LHyp (K)$
	cdlemk.t	$⊢ T = LTrn (K) (W)$
	cdlemk.r	$⊢ R = trL (K) (W)$
	cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
Assertion	cdlemk5	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to ((P \underset{˙}{\lor} N (P)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$

Proof

Step	Hyp	Ref	Expression
1		cdlemk.b	$⊢ B = {Base}_{K}$
2		cdlemk.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemk.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemk.a	$⊢ A = Atoms (K)$
5		cdlemk.h	$⊢ H = LHyp (K)$
6		cdlemk.t	$⊢ T = LTrn (K) (W)$
7		cdlemk.r	$⊢ R = trL (K) (W)$
8		cdlemk.m	$⊢ \underset{˙}{\land} = meet (K)$
9		simp11l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to K \in HL$
10		simp11r	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to W \in H$
11		simp12	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to F \in T$
12		simp21l	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to N \in T$
13		simp23	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to R (F) = R (N)$
14		simp22	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
15	1 2 3 4 5 6 7	cdlemk1	$⊢ ((K \in HL \land W \in H) \land (F \in T \land N \in T) \land (R (F) = R (N) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to P \underset{˙}{\lor} N (P) = F (P) \underset{˙}{\lor} R (F)$
16	9 10 11 12 13 14 15	syl222anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to P \underset{˙}{\lor} N (P) = F (P) \underset{˙}{\lor} R (F)$
17		simp13	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to G \in T$
18	1 2 3 4 5 6 7	cdlemk2	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W)) \to G (P) \underset{˙}{\lor} R (G \circ F^{-1}) = F (P) \underset{˙}{\lor} R (G \circ F^{-1})$
19	9 10 11 17 14 18	syl221anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to G (P) \underset{˙}{\lor} R (G \circ F^{-1}) = F (P) \underset{˙}{\lor} R (G \circ F^{-1})$
20	16 19	oveq12d	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (P \underset{˙}{\lor} N (P)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} R (G \circ F^{-1})) = (F (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G \circ F^{-1}))$
21		simp21r	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to X \in T$
22		simp33	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to R (G) \neq R (F)$
23		simp31	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to F \neq {I ↾}_{B}$
24		simp32	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to G \neq {I ↾}_{B}$
25	23 24	jca	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B})$
26	1 2 3 4 5 6 7 8	cdlemk5a	$⊢ ((K \in HL \land W \in H) \land (F \in T \land G \in T \land X \in T) \land (R (G) \neq R (F) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B}) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ((F (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$
27	9 10 11 17 21 22 25 14 26	syl233anc	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to ((F (P) \underset{˙}{\lor} R (F)) \underset{˙}{\land} (F (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$
28	20 27	eqbrtrd	$⊢ (((K \in HL \land W \in H) \land F \in T \land G \in T) \land ((N \in T \land X \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land R (F) = R (N)) \land (F \neq {I ↾}_{B} \land G \neq {I ↾}_{B} \land R (G) \neq R (F))) \to ((P \underset{˙}{\lor} N (P)) \underset{˙}{\land} (G (P) \underset{˙}{\lor} R (G \circ F^{-1}))) \underset{˙}{\leq} (X (P) \underset{˙}{\lor} R (X \circ F^{-1}))$

Metamath Proof Explorer

Theorem cdlemk5

Proof