Theorem cdlemn10 38336

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

Hypotheses

Ref	Expression
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

Ref	Expression
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 of Theorem cdlemn10

Step	Hyp	Ref	Expression
1		cdlemn10.b	. 2 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemn10.l	. 2 ⊢ ≤ = (le‘𝐾)
3		simp1l 1193	. . 3 ⊢ (((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)
4	3	hllatd 36494	. 2 ⊢ (((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)
5		simp22l 1288	. . 3 ⊢ (((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)
6		cdlemn10.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7	1, 6	atbase 36419	. . 3 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐵)
8	5, 7	syl 17	. 2 ⊢ (((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)
9		simp21l 1286	. . 3 ⊢ (((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)
10		cdlemn10.j	. . . 4 ⊢ ∨ = (join‘𝐾)
11	1, 10, 6	hlatjcl 36497	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑄 ∨ 𝑆) ∈ 𝐵)
12	3, 9, 5, 11	syl3anc 1367	. 2 ⊢ (((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)
13	1, 6	atbase 36419	. . . 4 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
14	9, 13	syl 17	. . 3 ⊢ (((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)
15		simp23l 1290	. . 3 ⊢ (((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)
16	1, 10	latjcl 17655	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) ∈ 𝐵)
17	4, 14, 15, 16	syl3anc 1367	. 2 ⊢ (((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)
18	2, 10, 6	hlatlej2 36506	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑆 ≤ (𝑄 ∨ 𝑆))
19	3, 9, 5, 18	syl3anc 1367	. 2 ⊢ (((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))
20		simp1r 1194	. . . . . 6 ⊢ (((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)
21		cdlemn10.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
22	1, 21	lhpbase 37128	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
23	20, 22	syl 17	. . . . 5 ⊢ (((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)
24	2, 10, 6	hlatlej1 36505	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑆))
25	3, 9, 5, 24	syl3anc 1367	. . . . 5 ⊢ (((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))
26		eqid 2821	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
27	1, 2, 10, 26, 6	atmod3i1 36994	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑄 ∨ 𝑆) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝑄 ≤ (𝑄 ∨ 𝑆)) → (𝑄 ∨ ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊)) = ((𝑄 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑊)))
28	3, 9, 12, 23, 25, 27	syl131anc 1379	. . . 4 ⊢ (((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)))
29		simp1 1132	. . . . . 6 ⊢ (((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))
30		simp21 1202	. . . . . 6 ⊢ (((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))
31		eqid 2821	. . . . . . 7 ⊢ (1.‘𝐾) = (1.‘𝐾)
32	2, 10, 31, 6, 21	lhpjat2 37151	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∨ 𝑊) = (1.‘𝐾))
33	29, 30, 32	syl2anc 586	. . . . 5 ⊢ (((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))
34	33	oveq2d 7166	. . . 4 ⊢ (((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)))
35		hlol 36491	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
36	3, 35	syl 17	. . . . 5 ⊢ (((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)
37	1, 26, 31	olm11 36357	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑄 ∨ 𝑆) ∈ 𝐵) → ((𝑄 ∨ 𝑆)(meet‘𝐾)(1.‘𝐾)) = (𝑄 ∨ 𝑆))
38	36, 12, 37	syl2anc 586	. . . 4 ⊢ (((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))
39	28, 34, 38	3eqtrrd 2861	. . 3 ⊢ (((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)))
40		simp31 1205	. . . . . . 7 ⊢ (((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)
41		cdlemn10.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
42		cdlemn10.r	. . . . . . . 8 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
43	2, 10, 26, 6, 21, 41, 42	trlval2 37293	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝑔) = ((𝑄 ∨ (𝑔‘𝑄))(meet‘𝐾)𝑊))
44	29, 40, 30, 43	syl3anc 1367	. . . . . 6 ⊢ (((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 1206	. . . . . . . 8 ⊢ (((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 7166	. . . . . . 7 ⊢ (((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 7165	. . . . . 6 ⊢ (((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 2856	. . . . 5 ⊢ (((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 1207	. . . . 5 ⊢ (((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 5083	. . . 4 ⊢ (((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, 26	latmcl 17656	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑆) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑄 ∨ 𝑆)(meet‘𝐾)𝑊) ∈ 𝐵)
52	4, 12, 23, 51	syl3anc 1367	. . . . 5 ⊢ (((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, 10	latjlej2 17670	. . . . 5 ⊢ ((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	4, 52, 15, 14, 53	syl13anc 1368	. . . 4 ⊢ (((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 15	. . 3 ⊢ (((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	39, 55	eqbrtrd 5081	. 2 ⊢ (((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, 4, 8, 12, 17, 19, 56	lattrd 17662	1 ⊢ (((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))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   class class class wbr 5059  ‘cfv 6350  (class class class)co 7150  Basecbs 16477  lecple 16566  joincjn 17548  meetcmee 17549  1.cp1 17642  Latclat 17649  OLcol 36304  Atomscatm 36393  HLchlt 36480  LHypclh 37114  LTrncltrn 37231  trLctrl 37288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-proset 17532  df-poset 17550  df-plt 17562  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-p0 17643  df-p1 17644  df-lat 17650  df-clat 17712  df-oposet 36306  df-ol 36308  df-oml 36309  df-covers 36396  df-ats 36397  df-atl 36428  df-cvlat 36452  df-hlat 36481  df-psubsp 36633  df-pmap 36634  df-padd 36926  df-lhyp 37118  df-laut 37119  df-ldil 37234  df-ltrn 37235  df-trl 37289

This theorem is referenced by:  cdlemn11pre  38340