cdlemeg47rv2

Description: Value of g_s(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO: FIX COMMENT. (Contributed by NM, 1-Apr-2013)

	Ref	Expression
Hypotheses	cdlemef47.b	$⊢ B = {Base}_{K}$
	cdlemef47.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemef47.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemef47.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemef47.a	$⊢ A = Atoms (K)$
	cdlemef47.h	$⊢ H = LHyp (K)$
	cdlemef47.v	$⊢ V = (Q \underset{˙}{\lor} P) \underset{˙}{\land} W$
	cdlemef47.n	$⊢ N = (v \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))$
	cdlemefs47.o	$⊢ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (N \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W))$
	cdlemef47.g	$⊢ G = (a \in B ⟼ if ((Q \neq P \land \neg a \underset{˙}{\leq} W), (ι c \in B \| \forall u \in A ((\neg u \underset{˙}{\leq} W \land u \underset{˙}{\lor} (a \underset{˙}{\land} W) = a) \to c = if (u \underset{˙}{\leq} (Q \underset{˙}{\lor} P), (ι b \in B \| \forall v \in A ((\neg v \underset{˙}{\leq} W \land \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \to b = O)), ⦋ u / v ⦌ N) \underset{˙}{\lor} (a \underset{˙}{\land} W))), a))$
Assertion	cdlemeg47rv2	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (R) = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (G (S) \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$

Proof

Step	Hyp	Ref	Expression
1		cdlemef47.b	$⊢ B = {Base}_{K}$
2		cdlemef47.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdlemef47.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdlemef47.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdlemef47.a	$⊢ A = Atoms (K)$
6		cdlemef47.h	$⊢ H = LHyp (K)$
7		cdlemef47.v	$⊢ V = (Q \underset{˙}{\lor} P) \underset{˙}{\land} W$
8		cdlemef47.n	$⊢ N = (v \underset{˙}{\lor} V) \underset{˙}{\land} (P \underset{˙}{\lor} ((Q \underset{˙}{\lor} v) \underset{˙}{\land} W))$
9		cdlemefs47.o	$⊢ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (N \underset{˙}{\lor} ((u \underset{˙}{\lor} v) \underset{˙}{\land} W))$
10		cdlemef47.g	$⊢ G = (a \in B ⟼ if ((Q \neq P \land \neg a \underset{˙}{\leq} W), (ι c \in B \| \forall u \in A ((\neg u \underset{˙}{\leq} W \land u \underset{˙}{\lor} (a \underset{˙}{\land} W) = a) \to c = if (u \underset{˙}{\leq} (Q \underset{˙}{\lor} P), (ι b \in B \| \forall v \in A ((\neg v \underset{˙}{\leq} W \land \neg v \underset{˙}{\leq} (Q \underset{˙}{\lor} P)) \to b = O)), ⦋ u / v ⦌ N) \underset{˙}{\lor} (a \underset{˙}{\land} W))), a))$
11	1 2 3 4 5 6 7 8 9 10	cdlemeg47rv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (R) = ⦋ R / u ⦌ ⦋ S / v ⦌ O$
12		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \in A$
13		nfcvd	$⊢ R \in A \to \underline{Ⅎ} u ((Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)))$
14		oveq1	$⊢ u = R \to u \underset{˙}{\lor} S = R \underset{˙}{\lor} S$
15	14	oveq1d	$⊢ u = R \to (u \underset{˙}{\lor} S) \underset{˙}{\land} W = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
16	15	oveq2d	$⊢ u = R \to ⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W) = ⦋ S / v ⦌ N \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)$
17	16	oveq2d	$⊢ u = R \to (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W)) = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
18	13 17	csbiegf	$⊢ R \in A \to ⦋ R / u ⦌ ((Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W))) = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
19	12 18	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ⦋ R / u ⦌ ((Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W))) = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
20		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \in A$
21		eqid	$⊢ (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W)) = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W))$
22	9 21	cdleme31se2	$⊢ S \in A \to ⦋ S / v ⦌ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W))$
23	20 22	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ⦋ S / v ⦌ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W))$
24	23	csbeq2dv	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ⦋ R / u ⦌ ⦋ S / v ⦌ O = ⦋ R / u ⦌ ((Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((u \underset{˙}{\lor} S) \underset{˙}{\land} W)))$
25		simp1	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
26		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
27		simp23	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
28		simp3r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
29	1 2 3 4 5 6 7 8 9 10	cdlemeg47b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to G (S) = ⦋ S / v ⦌ N$
30	25 26 27 28 29	syl121anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (S) = ⦋ S / v ⦌ N$
31	30	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (S) \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) = ⦋ S / v ⦌ N \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)$
32	31	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (Q \underset{˙}{\lor} P) \underset{˙}{\land} (G (S) \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (⦋ S / v ⦌ N \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
33	19 24 32	3eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ⦋ R / u ⦌ ⦋ S / v ⦌ O = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (G (S) \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
34	11 33	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G (R) = (Q \underset{˙}{\lor} P) \underset{˙}{\land} (G (S) \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$

Metamath Proof Explorer

Theorem cdlemeg47rv2

Proof