cdleme32f

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 20-Feb-2013)

	Ref	Expression
Hypotheses	cdleme32.b	$⊢ B = {Base}_{K}$
	cdleme32.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme32.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme32.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme32.a	$⊢ A = Atoms (K)$
	cdleme32.h	$⊢ H = LHyp (K)$
	cdleme32.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme32.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme32.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme32.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
	cdleme32.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
	cdleme32.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
	cdleme32.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
	cdleme32.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
Assertion	cdleme32f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to F (X) \underset{˙}{\leq} F (Y)$

Proof

Step	Hyp	Ref	Expression
1		cdleme32.b	$⊢ B = {Base}_{K}$
2		cdleme32.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme32.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme32.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme32.a	$⊢ A = Atoms (K)$
6		cdleme32.h	$⊢ H = LHyp (K)$
7		cdleme32.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme32.c	$⊢ C = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme32.d	$⊢ D = (t \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} t) \underset{˙}{\land} W))$
10		cdleme32.e	$⊢ E = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (D \underset{˙}{\lor} ((s \underset{˙}{\lor} t) \underset{˙}{\land} W))$
11		cdleme32.i	$⊢ I = (ι y \in B \| \forall t \in A ((\neg t \underset{˙}{\leq} W \land \neg t \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to y = E))$
12		cdleme32.n	$⊢ N = if (s \underset{˙}{\leq} (P \underset{˙}{\lor} Q), I, C)$
13		cdleme32.o	$⊢ O = (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
14		cdleme32.f	$⊢ F = (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
15		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to (K \in HL \land W \in H)$
16		simp21r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to Y \in B$
17		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to \neg Y \underset{˙}{\leq} W$
18	1 2 3 4 5 6	lhpmcvr2	$⊢ ((K \in HL \land W \in H) \land (Y \in B \land \neg Y \underset{˙}{\leq} W)) \to \exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y)$
19	15 16 17 18	syl12anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to \exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y)$
20		nfv	$⊢ Ⅎ s (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y)$
21		nfcv	$⊢ \underline{Ⅎ} s B$
22		nfv	$⊢ Ⅎ s (P \neq Q \land \neg x \underset{˙}{\leq} W)$
23		nfra1	$⊢ Ⅎ s \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W))$
24	23 21	nfriota	$⊢ \underline{Ⅎ} s (ι z \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (x \underset{˙}{\land} W) = x) \to z = N \underset{˙}{\lor} (x \underset{˙}{\land} W)))$
25	13 24	nfcxfr	$⊢ \underline{Ⅎ} s O$
26		nfcv	$⊢ \underline{Ⅎ} s x$
27	22 25 26	nfif	$⊢ \underline{Ⅎ} s if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x)$
28	21 27	nfmpt	$⊢ \underline{Ⅎ} s (x \in B ⟼ if ((P \neq Q \land \neg x \underset{˙}{\leq} W), O, x))$
29	14 28	nfcxfr	$⊢ \underline{Ⅎ} s F$
30		nfcv	$⊢ \underline{Ⅎ} s X$
31	29 30	nffv	$⊢ \underline{Ⅎ} s F (X)$
32		nfcv	$⊢ \underline{Ⅎ} s \underset{˙}{\leq}$
33		nfcv	$⊢ \underline{Ⅎ} s Y$
34	29 33	nffv	$⊢ \underline{Ⅎ} s F (Y)$
35	31 32 34	nfbr	$⊢ Ⅎ s F (X) \underset{˙}{\leq} F (Y)$
36		simpl1	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
37		simpl2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y))) \to ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W))$
38		simprl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y))) \to s \in A$
39		simprrl	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y))) \to \neg s \underset{˙}{\leq} W$
40	38 39	jca	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
41		simprrr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y))) \to s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y$
42		simpl3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y))) \to X \underset{˙}{\leq} Y$
43	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y \land X \underset{˙}{\leq} Y)) \to F (X) \underset{˙}{\leq} F (Y)$
44	36 37 40 41 42 43	syl113anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \land (s \in A \land (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y))) \to F (X) \underset{˙}{\leq} F (Y)$
45	44	exp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to (s \in A \to ((\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y) \to F (X) \underset{˙}{\leq} F (Y)))$
46	20 35 45	rexlimd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to (\exists s \in A (\neg s \underset{˙}{\leq} W \land s \underset{˙}{\lor} (Y \underset{˙}{\land} W) = Y) \to F (X) \underset{˙}{\leq} F (Y))$
47	19 46	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to F (X) \underset{˙}{\leq} F (Y)$

Metamath Proof Explorer

Theorem cdleme32f

Proof