cdlemn10

Description: Part of proof of Lemma N of Crawley p. 121 line 36. (Contributed by NM, 27-Feb-2014)

	Ref	Expression
Hypotheses	cdlemn10.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemn10.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemn10.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemn10.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemn10.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemn10.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemn10.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion	cdlemn10	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemn10.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemn10.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemn10.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemn10.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdlemn10.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdlemn10.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7		cdlemn10.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8		simp1l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝐾 ∈ HL )
9	8	hllatd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝐾 ∈ Lat )
10		simp22l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ∈ 𝐴 )
11	1 4	atbase	⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐵 )
12	10 11	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ∈ 𝐵 )
13		simp21l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑄 ∈ 𝐴 )
14	1 3 4	hlatjcl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 )
15	8 13 10 14	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 )
16	1 4	atbase	⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
17	13 16	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑄 ∈ 𝐵 )
18		simp23l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 )
19	1 3	latjcl	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑄 ∨ 𝑋 ) ∈ 𝐵 )
20	9 17 18 19	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑋 ) ∈ 𝐵 )
21	2 3 4	hlatlej2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → 𝑆 ≤ ( 𝑄 ∨ 𝑆 ) )
22	8 13 10 21	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ≤ ( 𝑄 ∨ 𝑆 ) )
23		simp1r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑊 ∈ 𝐻 )
24	1 5	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
25	23 24	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑊 ∈ 𝐵 )
26	2 3 4	hlatlej1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → 𝑄 ≤ ( 𝑄 ∨ 𝑆 ) )
27	8 13 10 26	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑄 ≤ ( 𝑄 ∨ 𝑆 ) )
28		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
29	1 2 3 28 4	atmod3i1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ∧ 𝑄 ≤ ( 𝑄 ∨ 𝑆 ) ) → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑊 ) ) )
30	8 13 15 25 27 29	syl131anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑊 ) ) )
31		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
32		simp21	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
33		eqid	⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
34	2 3 33 4 5	lhpjat2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑄 ∨ 𝑊 ) = ( 1. ‘ 𝐾 ) )
35	31 32 34	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑊 ) = ( 1. ‘ 𝐾 ) )
36	35	oveq2d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑊 ) ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) )
37		hlol	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
38	8 37	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝐾 ∈ OL )
39	1 28 33	olm11	⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑄 ∨ 𝑆 ) )
40	38 15 39	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑄 ∨ 𝑆 ) )
41	30 36 40	3eqtrrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
42		simp31	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑔 ∈ 𝑇 )
43	2 3 28 4 5 6 7	trlval2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑔 ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝑔 ) = ( ( 𝑄 ∨ ( 𝑔 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
44	31 42 32 43	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑔 ) = ( ( 𝑄 ∨ ( 𝑔 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
45		simp32	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑔 ‘ 𝑄 ) = 𝑆 )
46	45	oveq2d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ ( 𝑔 ‘ 𝑄 ) ) = ( 𝑄 ∨ 𝑆 ) )
47	46	oveq1d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ ( 𝑔 ‘ 𝑄 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) )
48	44 47	eqtrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑔 ) = ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) )
49		simp33	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 )
50	48 49	eqbrtrrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑋 )
51	1 28	latmcl	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ 𝑆 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
52	9 15 25 51	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
53	1 2 3	latjlej2	⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) ) → ( ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑋 → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ≤ ( 𝑄 ∨ 𝑋 ) ) )
54	9 52 18 17 53	syl13anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑋 → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ≤ ( 𝑄 ∨ 𝑋 ) ) )
55	50 54	mpd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ ( ( 𝑄 ∨ 𝑆 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ≤ ( 𝑄 ∨ 𝑋 ) )
56	41 55	eqbrtrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑆 ) ≤ ( 𝑄 ∨ 𝑋 ) )
57	1 2 9 12 15 20 22 56	lattrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝑔 ∈ 𝑇 ∧ ( 𝑔 ‘ 𝑄 ) = 𝑆 ∧ ( 𝑅 ‘ 𝑔 ) ≤ 𝑋 ) ) → 𝑆 ≤ ( 𝑄 ∨ 𝑋 ) )

Metamath Proof Explorer

Theorem cdlemn10

Proof