cdleme26f2ALTN

Description: Part of proof of Lemma E in Crawley p. 113. cdleme26fALTN with s and t swapped (this case is not mentioned by them). If s <_ t \/ v, then f(s) <_ f_s(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme26.b	$⊢ B = {Base}_{K}$
	cdleme26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme26.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme26.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme26.a	$⊢ A = Atoms (K)$
	cdleme26.h	$⊢ H = LHyp (K)$
	cdleme26f2.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme26f2.f	$⊢ G = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme26f2.n	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} ((T \underset{˙}{\lor} s) \underset{˙}{\land} W))$
	cdleme26f2.e	$⊢ E = (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = O))$
Assertion	cdleme26f2ALTN	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to G \underset{˙}{\leq} (E \underset{˙}{\lor} V)$

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	$⊢ B = {Base}_{K}$
2		cdleme26.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		cdleme26.j	$⊢ \underset{˙}{\lor} = join (K)$
4		cdleme26.m	$⊢ \underset{˙}{\land} = meet (K)$
5		cdleme26.a	$⊢ A = Atoms (K)$
6		cdleme26.h	$⊢ H = LHyp (K)$
7		cdleme26f2.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
8		cdleme26f2.f	$⊢ G = (s \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} s) \underset{˙}{\land} W))$
9		cdleme26f2.n	$⊢ O = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (G \underset{˙}{\lor} ((T \underset{˙}{\lor} s) \underset{˙}{\land} W))$
10		cdleme26f2.e	$⊢ E = (ι u \in B \| \forall s \in A ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to u = O))$
11		simp11	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
12		simp23	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (T \in A \land \neg T \underset{˙}{\leq} W)$
13		simp31r	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
14		simp12r	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
15		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to P \neq Q$
16	13 14 15	3jca	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)$
17		simp21	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
18		simp22	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
19		simp13	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (s \in A \land \neg s \underset{˙}{\leq} W)$
20		simp32	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V))$
21		simp33	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (V \in A \land V \underset{˙}{\leq} W)$
22	2 3 4 5 6 7 8 9	cdleme22f2	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((s \in A \land \neg s \underset{˙}{\leq} W) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to G \underset{˙}{\leq} (O \underset{˙}{\lor} V)$
23	11 12 16 17 18 19 20 21 22	syl323anc	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to G \underset{˙}{\leq} (O \underset{˙}{\lor} V)$
24		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to T \in A$
25		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to \neg T \underset{˙}{\leq} W$
26	1 2 3 4 5 6 7 8 9 10	cdleme25cl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to E \in B$
27	11 17 18 24 25 15 14 26	syl322anc	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to E \in B$
28		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to s \in A$
29		simp31	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
30	1	fvexi	$⊢ B \in V$
31	30 10	riotasv	$⊢ (E \in B \land s \in A \land (\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to E = O$
32	27 28 29 31	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to E = O$
33	32	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to E \underset{˙}{\lor} V = O \underset{˙}{\lor} V$
34	23 33	breqtrrd	$⊢ (((K \in HL \land W \in H) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \in A \land \neg s \underset{˙}{\leq} W)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land ((\neg s \underset{˙}{\leq} W \land \neg s \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \land (s \neq T \land s \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to G \underset{˙}{\leq} (E \underset{˙}{\lor} V)$

Metamath Proof Explorer

Theorem cdleme26f2ALTN

Proof