cdleme32le

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	cdleme32le	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \to F (X) \underset{˙}{\leq} F (Y)$

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		simpl1	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
16		simpl2l	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to X \in B$
17		simpl2r	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to Y \in B$
18		simpr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to (P \neq Q \land \neg X \underset{˙}{\leq} W)$
19		simpl3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to X \underset{˙}{\leq} Y$
20	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to F (X) \underset{˙}{\leq} F (Y)$
21	15 16 17 18 19 20	syl131anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) \underset{˙}{\leq} F (Y)$
22		simp11	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
23		simp12	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to (X \in B \land Y \in B)$
24		simp3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)$
25		simp2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to (P \neq Q \land \neg Y \underset{˙}{\leq} W)$
26		simp13	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to X \underset{˙}{\leq} Y$
27	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((X \in B \land Y \in B) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \land X \underset{˙}{\leq} Y) \to F (X) \underset{˙}{\leq} F (Y)$
28	22 23 24 25 26 27	syl131anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) \underset{˙}{\leq} F (Y)$
29	28	3exp	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \to ((P \neq Q \land \neg Y \underset{˙}{\leq} W) \to (\neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \to F (X) \underset{˙}{\leq} F (Y)))$
30		simp13	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to X \underset{˙}{\leq} Y$
31		simp12l	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to X \in B$
32		simp3	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)$
33	14	cdleme31fv2	$⊢ (X \in B \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) = X$
34	31 32 33	syl2anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) = X$
35		simp12r	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to Y \in B$
36		simp2	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W)$
37	14	cdleme31fv2	$⊢ (Y \in B \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W)) \to F (Y) = Y$
38	35 36 37	syl2anc	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (Y) = Y$
39	30 34 38	3brtr4d	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) \underset{˙}{\leq} F (Y)$
40	39	3exp	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \to (\neg (P \neq Q \land \neg Y \underset{˙}{\leq} W) \to (\neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \to F (X) \underset{˙}{\leq} F (Y)))$
41	29 40	pm2.61d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \to (\neg (P \neq Q \land \neg X \underset{˙}{\leq} W) \to F (X) \underset{˙}{\leq} F (Y))$
42	41	imp	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \land \neg (P \neq Q \land \neg X \underset{˙}{\leq} W)) \to F (X) \underset{˙}{\leq} F (Y)$
43	21 42	pm2.61dan	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (X \in B \land Y \in B) \land X \underset{˙}{\leq} Y) \to F (X) \underset{˙}{\leq} F (Y)$

Metamath Proof Explorer

Theorem cdleme32le

Proof