cdleme48fvg

Description: Remove P =/= Q condition in cdleme48fv . TODO: Can this replace uses of cdleme32a ? TODO: Can this be used to help prove the R or S case where X is an atom? TODO: Can this be proved more directly by eliminating P =/= Q in earlier theorems? Should this replace uses of cdleme48fv ? (Contributed by NM, 23-Apr-2013)

	Ref	Expression
Hypotheses	cdlemef46.b	$⊢ B = {Base}_{K}$
	cdlemef46.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemef46.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemef46.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemef46.a	$⊢ A = Atoms (K)$
	cdlemef46.h	$⊢ H = LHyp (K)$
	cdlemef46.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdlemef46.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemefs46.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdlemef46.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
Assertion	cdleme48fvg	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) = F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$

Proof

Step	Hyp	Ref	Expression
1		cdlemef46.b	$⊢ B = {Base}_{K}$
2		cdlemef46.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemef46.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemef46.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemef46.a	$⊢ A = Atoms (K)$
6		cdlemef46.h	$⊢ H = LHyp (K)$
7		cdlemef46.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdlemef46.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
9		cdlemefs46.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdlemef46.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E)), ⦋ s / t ⦌ D) \underset{˙}{\lor} (x \underset{˙}{\land} W))), x))$
11		simpl3r	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P = Q) \to S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X$
12		simp3ll	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to S \in A$
13	12	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P = Q) \to S \in A$
14	1 5	atbase	$⊢ S \in A \to S \in B$
15	13 14	syl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P = Q) \to S \in B$
16	10	cdleme31id	$⊢ (S \in B \land P = Q) \to F (S) = S$
17	15 16	sylancom	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P = Q) \to F (S) = S$
18	17	oveq1d	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P = Q) \to F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W) = S \underset{˙}{\lor} (X \underset{˙}{\land} W)$
19		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to X \in B$
20	10	cdleme31id	$⊢ (X \in B \land P = Q) \to F (X) = X$
21	19 20	sylan	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P = Q) \to F (X) = X$
22	11 18 21	3eqtr4rd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P = Q) \to F (X) = F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$
23		simpl1	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P \neq Q) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
24		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P \neq Q) \to P \neq Q$
25		simpl2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P \neq Q) \to (X \in B \land \neg X \underset{˙}{\leq} W)$
26		simpl3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P \neq Q) \to ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)$
27	1 2 3 4 5 6 7 8 9 10	cdleme48fv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (X \in B \land \neg X \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) = F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$
28	23 24 25 26 27	syl121anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \land P \neq Q) \to F (X) = F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$
29	22 28	pm2.61dane	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land \neg X \underset{˙}{\leq} W) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land S \underset{˙}{\lor} (X \underset{˙}{\land} W) = X)) \to F (X) = F (S) \underset{˙}{\lor} (X \underset{˙}{\land} W)$

Metamath Proof Explorer

Theorem cdleme48fvg

Proof