Theorem cdlemg4c 40988

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 767	. . . . . . . 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 1218	. . . . . . . 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 1219	. . . . . . . 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 40986	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑄) ∨ 𝑉) = (𝑄 ∨ (𝐺‘𝑄)))
12	1, 2, 3, 11	syl3anc 1374	. . . . . . 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 484	. . . . . . . 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 767	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝐾 ∈ HL)
15	14	hllatd 39740	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝐾 ∈ Lat)
16		simpr1l 1232	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑃 ∈ 𝐴)
17		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 5	atbase 39665	. . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		simpr3 1198	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝐺 ∈ 𝑇)
22	17, 6, 7, 8	trlcl 40540	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
23	20, 21, 22	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (𝑅‘𝐺) ∈ (Base‘𝐾))
24	10, 23	eqeltrid 2841	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑉 ∈ (Base‘𝐾))
25	17, 4, 9	latlej2 18384	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 ≤ (𝑃 ∨ 𝑉))
26	15, 19, 24, 25	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑉 ≤ (𝑃 ∨ 𝑉))
27	26	adantr 480	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑉))
28		simpr2l 1234	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → 𝑄 ∈ 𝐴)
29	17, 5	atbase 39665	. . . . . . . . . . . 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 40501	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐺‘𝑄) ∈ (Base‘𝐾))
32	20, 21, 30, 31	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (𝐺‘𝑄) ∈ (Base‘𝐾))
33	17, 9	latjcl 18374	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
34	15, 19, 24, 33	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
35	17, 4, 9	latjle12 18385	. . . . . . . . . 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 1375	. . . . . . . . 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 480	. . . . . . . 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 713	. . . . . . 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 5124	. . . . . 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 480	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇)) ∧ (𝐺‘𝑄) ≤ (𝑃 ∨ 𝑉)) → 𝐾 ∈ Lat)
41	30	adantr 480	. . . . . . 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 480	. . . . . . 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 480	. . . . . . . 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 585	. . . . . . . . 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 2841	. . . . . . . 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 1374	. . . . . . 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 18385	. . . . . . 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 1375	. . . . . 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 257	. . . . 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 494	. . . 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 412	. . 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 1118	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  Latclat 18366  Atomscatm 39639  HLchlt 39726  LHypclh 40360  LTrncltrn 40477  trLctrl 40534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-p1 18359  df-lat 18367  df-clat 18434  df-oposet 39552  df-ol 39554  df-oml 39555  df-covers 39642  df-ats 39643  df-atl 39674  df-cvlat 39698  df-hlat 39727  df-psubsp 39879  df-pmap 39880  df-padd 40172  df-lhyp 40364  df-laut 40365  df-ldil 40480  df-ltrn 40481  df-trl 40535

This theorem is referenced by:  cdlemg4d  40989