cdlemkfid3N

Description: TODO: is this useful or should it be deleted? (Contributed by NM, 29-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	$⊢ Y = (P \underset{˙}{\lor} R (g)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (g \circ b^{-1}))$
Assertion	cdlemkfid3N	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ⦋ G / g ⦌ Y = G (P)$

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		simp22	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to G \in T$
12	10	cdlemk41	$⊢ G \in T \to ⦋ G / g ⦌ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
13	11 12	syl	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ⦋ G / g ⦌ Y = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1}))$
14		simp1	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ((K \in HL \land W \in H) \land F = N)$
15		simp21l	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \in T$
16		simp21r	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to F \neq {I ↾}_{B}$
17		simp23l	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b \in T$
18		simp31	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (b) \neq R (F)$
19		simp33	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
20	1 2 3 4 5 6 7 8 9	cdlemkfid2N	$⊢ (((K \in HL \land W \in H) \land F = N) \land (F \in T \land F \neq {I ↾}_{B} \land b \in T) \land (R (b) \neq R (F) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to Z = b (P)$
21	14 15 16 17 18 19 20	syl132anc	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to Z = b (P)$
22	21	oveq1d	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to Z \underset{˙}{\lor} R (G \circ b^{-1}) = b (P) \underset{˙}{\lor} R (G \circ b^{-1})$
23	22	oveq2d	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (Z \underset{˙}{\lor} R (G \circ b^{-1})) = (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (b (P) \underset{˙}{\lor} R (G \circ b^{-1}))$
24		simp1l	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
25		simp23r	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to b \neq {I ↾}_{B}$
26		simp32	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (b) \neq R (G)$
27	26	necomd	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to R (G) \neq R (b)$
28	1 2 3 4 5 6 7 8	cdlemkfid1N	$⊢ ((K \in HL \land W \in H) \land (b \in T \land b \neq {I ↾}_{B} \land G \in T) \land (R (G) \neq R (b) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (b (P) \underset{˙}{\lor} R (G \circ b^{-1})) = G (P)$
29	24 17 25 11 27 19 28	syl132anc	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to (P \underset{˙}{\lor} R (G)) \underset{˙}{\land} (b (P) \underset{˙}{\lor} R (G \circ b^{-1})) = G (P)$
30	13 23 29	3eqtrd	$⊢ (((K \in HL \land W \in H) \land F = N) \land ((F \in T \land F \neq {I ↾}_{B}) \land G \in T \land (b \in T \land b \neq {I ↾}_{B})) \land (R (b) \neq R (F) \land R (b) \neq R (G) \land (P \in A \land \neg P \underset{˙}{\leq} W))) \to ⦋ G / g ⦌ Y = G (P)$

Metamath Proof Explorer

Theorem cdlemkfid3N

Proof