Theorem cdlemg4c 41104

Description: TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013.)

Hypotheses

Ref	Expression
cdlemg4.l	⊢ ≤ = (le‘𝐾)
cdlemg4.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg4.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg4.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg4.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemg4.j	⊢ ∨ = (join‘𝐾)
cdlemg4b.v	⊢ 𝑉 = (𝑅‘𝐺)

Assertion

Ref	Expression
cdlemg4c	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉)) → ¬ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉))

Proof of Theorem cdlemg4c

Step	Hyp	Ref	Expression
1		simpll 772	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simplr2 1223	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
3		simplr3 1224	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝐺 ∈ 𝑇)
4		cdlemg4.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
5		cdlemg4.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg4.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemg4.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemg4.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
9		cdlemg4.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
10		cdlemg4b.v	. . . . . . . . 9 ⊢ 𝑉 = (𝑅‘𝐺)
11	4, 5, 6, 7, 8, 9, 10	cdlemg4b2 41102	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑄) ∨ 𝑉) = (𝑄 ∨ (𝐺‘𝑄)))
12	1, 2, 3, 11	syl3anc 1379	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → ((𝐺‘𝑄) ∨ 𝑉) = (𝑄 ∨ (𝐺‘𝑄)))
13		simpr 485	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉))
14		simpll 772	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝐾 ∈ HL)
15	14	hllatd 39856	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝐾 ∈ Lat)
16		simpr1l 1237	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑃 ∈ 𝐴)
17		eqid 2739	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 5	atbase 39781	. . . . . . . . . . 11 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
19	16, 18	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑃 ∈ (Base‘𝐾))
20		simpl 483	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		simpr3 1203	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝐺 ∈ 𝑇)
22	17, 6, 7, 8	trlcl 40656	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
23	20, 21, 22	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (𝑅‘𝐺) ∈ (Base‘𝐾))
24	10, 23	eqeltrid 2843	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑉 ∈ (Base‘𝐾))
25	17, 4, 9	latlej2 18406	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 ≤ (𝑃 ∨ 𝑉))
26	15, 19, 24, 25	syl3anc 1379	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑉 ≤ (𝑃 ∨ 𝑉))
27	26	adantr 481	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑉))
28		simpr2l 1239	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑄 ∈ 𝐴)
29	17, 5	atbase 39781	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
30	28, 29	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑄 ∈ (Base‘𝐾))
31	17, 6, 7	ltrncl 40617	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐺‘𝑄) ∈ (Base‘𝐾))
32	20, 21, 30, 31	syl3anc 1379	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (𝐺‘𝑄) ∈ (Base‘𝐾))
33	17, 9	latjcl 18396	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
34	15, 19, 24, 33	syl3anc 1379	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
35	17, 4, 9	latjle12 18407	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺‘𝑄) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → (((𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉) ∧ 𝑉 ≤ (𝑃 ∨ 𝑉)) ↔ ((𝐺‘𝑄) ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)))
36	15, 32, 24, 34, 35	syl13anc 1380	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (((𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉) ∧ 𝑉 ≤ (𝑃 ∨ 𝑉)) ↔ ((𝐺‘𝑄) ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)))
37	36	adantr 481	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (((𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉) ∧ 𝑉 ≤ (𝑃 ∨ 𝑉)) ↔ ((𝐺‘𝑄) ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)))
38	13, 27, 37	mpbi2and 718	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → ((𝐺‘𝑄) ∨ 𝑉) ≤ (𝑃 ∨ 𝑉))
39	12, 38	eqbrtrrd 5096	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (𝑄 ∨ (𝐺‘𝑄)) ≤ (𝑃 ∨ 𝑉))
40	15	adantr 481	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝐾 ∈ Lat)
41	30	adantr 481	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝑄 ∈ (Base‘𝐾))
42	32	adantr 481	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (𝐺‘𝑄) ∈ (Base‘𝐾))
43	19	adantr 481	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝑃 ∈ (Base‘𝐾))
44	1, 3, 22	syl2anc 590	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (𝑅‘𝐺) ∈ (Base‘𝐾))
45	10, 44	eqeltrid 2843	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝑉 ∈ (Base‘𝐾))
46	40, 43, 45, 33	syl3anc 1379	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
47	17, 4, 9	latjle12 18407	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝐺‘𝑄) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) ↔ (𝑄 ∨ (𝐺‘𝑄)) ≤ (𝑃 ∨ 𝑉)))
48	40, 41, 42, 46, 47	syl13anc 1380	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → ((𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) ↔ (𝑄 ∨ (𝐺‘𝑄)) ≤ (𝑃 ∨ 𝑉)))
49	39, 48	mpbird 258	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → (𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)))
50	49	simpld 495	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝑄 ≤ (𝑃 ∨ 𝑉))
51	50	ex 413	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → ((𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉) → 𝑄 ≤ (𝑃 ∨ 𝑉)))
52	51	con3d 152	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (¬ 𝑄 ≤ (𝑃 ∨ 𝑉) → ¬ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)))
53	52	3impia 1123	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑉)) → ¬ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  lecple 17218  joincjn 18268  Latclat 18388  Atomscatm 39755  HLchlt 39842  LHypclh 40476  LTrncltrn 40593  trLctrl 40650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18389  df-clat 18456  df-oposet 39668  df-ol 39670  df-oml 39671  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-psubsp 39995  df-pmap 39996  df-padd 40288  df-lhyp 40480  df-laut 40481  df-ldil 40596  df-ltrn 40597  df-trl 40651

This theorem is referenced by:  cdlemg4d  41105