cdlemg18b

Description: Lemma for cdlemg18c . TODO: fix comment. (Contributed by NM, 15-May-2013)

	Ref	Expression
Hypotheses	cdlemg12.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemg12.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemg12.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemg12.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemg12.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemg12.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemg12b.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	cdlemg18b.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
Assertion	cdlemg18b	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdlemg12.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdlemg12.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdlemg12.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdlemg12.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdlemg12.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7		cdlemg12b.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8		cdlemg18b.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
9		simp33	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) )
10		simp3r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
11		simp1l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝐾 ∈ HL )
12		simp1r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑊 ∈ 𝐻 )
13		simp21	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
14		simp22l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑄 ∈ 𝐴 )
15		simp3l1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ≠ 𝑄 )
16	1 2 3 4 5 8	cdleme0a	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑈 ∈ 𝐴 )
17	11 12 13 14 15 16	syl212anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑈 ∈ 𝐴 )
18		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
19		simp23	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝐹 ∈ 𝑇 )
20	1 4 5 6	ltrnat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
21	18 19 14 20	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
22	1 2 4	hlatlej1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → 𝑈 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
23	11 17 21 22	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑈 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
24	11	hllatd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝐾 ∈ Lat )
25		simp21l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ∈ 𝐴 )
26		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
27	26 4	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
28	25 27	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
29	26 4	atbase	⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
30	17 29	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
31	26 2 4	hlatjcl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
32	11 17 21 31	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
33	26 1 2	latjle12	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ↔ ( 𝑃 ∨ 𝑈 ) ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
34	24 28 30 32 33	syl13anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ↔ ( 𝑃 ∨ 𝑈 ) ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
35	10 23 34	mpbi2and	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∨ 𝑈 ) ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
36	1 2 3 4 5 8	cdleme0cp	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) )
37	11 12 13 14 36	syl22anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) )
38		simp22	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
39	5 6 1 2 4 3 8	cdlemg2kq	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ 𝑈 ) )
40	18 13 38 19 39	syl121anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ 𝑈 ) )
41	2 4	hlatjcom	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑄 ) ∨ 𝑈 ) = ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
42	11 21 17 41	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ 𝑈 ) = ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
43	40 42	eqtr2d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )
44	35 37 43	3brtr3d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )
45	1 4 5 6	ltrnat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
46	18 19 25 45	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
47	1 2 4	ps-1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ↔ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
48	11 25 14 15 46 21 47	syl132anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ↔ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
49	44 48	mpbid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )
50	2 4	hlatjcom	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
51	11 46 21 50	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
52	49 51	eqtr2d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) )
53	52	3exp	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) ) ) )
54	53	exp4a	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) ) ) ) )
55	54	3imp	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) ) )
56	55	necon3ad	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) → ¬ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
57	9 56	mpd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )

Metamath Proof Explorer

Theorem cdlemg18b

Proof