cdlemg1a

Description: Shorter expression for G . TODO: fix comment. TODO: shorten using cdleme or vice-versa? Also, if not shortened with cdleme , then it can be moved up to save repeating hypotheses. (Contributed by NM, 15-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg1.b	$⊢ B = {Base}_{K}$
	cdlemg1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemg1.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemg1.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemg1.a	$⊢ A = Atoms (K)$
	cdlemg1.h	$⊢ H = LHyp (K)$
	cdlemg1.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdlemg1.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemg1.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemg1.g	$⊢ G = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
	cdlemg1.t	$⊢ T = LTrn (K) (W)$
Assertion	cdlemg1a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to G = (ι f \in T \| f (P) = Q)$

Proof

Step	Hyp	Ref	Expression
1		cdlemg1.b	$⊢ B = {Base}_{K}$
2		cdlemg1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemg1.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemg1.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemg1.a	$⊢ A = Atoms (K)$
6		cdlemg1.h	$⊢ H = LHyp (K)$
7		cdlemg1.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemg1.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemg1.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemg1.g	$⊢ G = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
11		cdlemg1.t	$⊢ T = LTrn (K) (W)$
12	1 2 3 4 5 6 7 8 9 10 11	cdleme50ltrn	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to G \in T$
13		simpll1	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \land f (P) = Q) \to (K \in HL \land W \in H)$
14		simplr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \land f (P) = Q) \to f \in T$
15	12	ad2antrr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \land f (P) = Q) \to G \in T$
16		simpll2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \land f (P) = Q) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
17		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \land f (P) = Q) \to f (P) = Q$
18	1 2 3 4 5 6 7 8 9 10	cdleme17d	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to G (P) = Q$
19	18	ad2antrr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \land f (P) = Q) \to G (P) = Q$
20	17 19	eqtr4d	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \land f (P) = Q) \to f (P) = G (P)$
21	2 5 6 11	cdlemd	$⊢ (((K \in HL \land W \in H) \land f \in T \land G \in T) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land f (P) = G (P)) \to f = G$
22	13 14 15 16 20 21	syl311anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \land f (P) = Q) \to f = G$
23	22	ex	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \to (f (P) = Q \to f = G)$
24	18	adantr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \to G (P) = Q$
25		fveq1	$⊢ f = G \to f (P) = G (P)$
26	25	eqeq1d	$⊢ f = G \to (f (P) = Q \leftrightarrow G (P) = Q)$
27	24 26	syl5ibrcom	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \to (f = G \to f (P) = Q)$
28	23 27	impbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land f \in T) \to (f (P) = Q \leftrightarrow f = G)$
29	12 28	riota5	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to (ι f \in T \| f (P) = Q) = G$
30	29	eqcomd	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to G = (ι f \in T \| f (P) = Q)$

Metamath Proof Explorer

Theorem cdlemg1a

Proof