cdleme17d3

	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	cdleme17d3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to F (P) = Q$

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		simpl1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (K \in HL \land W \in H)$
12		simpl2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
13		simpl3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
14		simpr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to P \neq Q$
15	2 3 5 6	cdlemb2	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to \exists e \in A (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
16	11 12 13 14 15	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to \exists e \in A (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
17		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (e \in A \land (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
18		simp2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (e \in A \land (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to P \neq Q$
19		simp3l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (e \in A \land (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to e \in A$
20		simp3rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (e \in A \land (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to \neg e \underset{˙}{\leq} W$
21	19 20	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (e \in A \land (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to (e \in A \land \neg e \underset{˙}{\leq} W)$
22		simp3rr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (e \in A \land (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
23	1 2 3 4 5 6 7 8 9 10	cdleme17d2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (e \in A \land \neg e \underset{˙}{\leq} W)) \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F (P) = Q$
24	17 18 21 22 23	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q \land (e \in A \land (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to F (P) = Q$
25	24	3expia	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to ((e \in A \land (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F (P) = Q)$
26	25	expd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (e \in A \to ((\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F (P) = Q))$
27	26	rexlimdv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to (\exists e \in A (\neg e \underset{˙}{\leq} W \land \neg e \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F (P) = Q)$
28	16 27	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land P \neq Q) \to F (P) = Q$

Metamath Proof Explorer

Theorem cdleme17d3

Proof