Theorem cdlemg6c 40880

Description: TODO: FIX COMMENT. (Contributed by NM, 27-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
cdlemg6c	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉)) → (𝐹‘(𝐺‘𝑄)) = 𝑄))

Distinct variable groups:   𝐴,𝑟   𝐹,𝑟   𝐺,𝑟   𝐻,𝑟   ∨ ,𝑟   𝐾,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑇,𝑟   𝑉,𝑟   𝑊,𝑟

Allowed substitution hint:   𝑅(𝑟)

Proof of Theorem cdlemg6c

Step	Hyp	Ref	Expression
1		simpl1 1192	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simprl 770	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊))
3		simpl22 1253	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
4		simpl23 1254	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝐹 ∈ 𝑇)
5		simpl31 1255	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝐺 ∈ 𝑇)
6		simprr 772	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))
7		simpl1l 1225	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝐾 ∈ HL)
8		simp22l 1293	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑄 ∈ 𝐴)
9	8	adantr 480	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑄 ∈ 𝐴)
10		simprll 778	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑟 ∈ 𝐴)
11		cdlemg4b.v	. . . . . . 7 ⊢ 𝑉 = (𝑅‘𝐺)
12		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
13		cdlemg4.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
14		cdlemg4.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
15		cdlemg4.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
16	12, 13, 14, 15	trlcl 40424	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
17	1, 5, 16	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑅‘𝐺) ∈ (Base‘𝐾))
18	11, 17	eqeltrid 2840	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑉 ∈ (Base‘𝐾))
19		simp22r 1294	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → ¬ 𝑄 ≤ 𝑊)
20	19	adantr 480	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ¬ 𝑄 ≤ 𝑊)
21		cdlemg4.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
22	21, 13, 14, 15	trlle 40444	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ≤ 𝑊)
23	1, 5, 22	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑅‘𝐺) ≤ 𝑊)
24	11, 23	eqbrtrid 5133	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑉 ≤ 𝑊)
25		simp1l 1198	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝐾 ∈ HL)
26	25	hllatd 39624	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝐾 ∈ Lat)
27	26	adantr 480	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝐾 ∈ Lat)
28		cdlemg4.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
29	12, 28	atbase 39549	. . . . . . . . . . 11 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
30	8, 29	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑄 ∈ (Base‘𝐾))
31	30	adantr 480	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑄 ∈ (Base‘𝐾))
32		simp1r 1199	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑊 ∈ 𝐻)
33	12, 13	lhpbase 40258	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑊 ∈ (Base‘𝐾))
35	34	adantr 480	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑊 ∈ (Base‘𝐾))
36	12, 21	lattr 18367	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊) → 𝑄 ≤ 𝑊))
37	27, 31, 18, 35, 36	syl13anc 1374	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ((𝑄 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊) → 𝑄 ≤ 𝑊))
38	24, 37	mpan2d 694	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ≤ 𝑉 → 𝑄 ≤ 𝑊))
39	20, 38	mtod 198	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ¬ 𝑄 ≤ 𝑉)
40		cdlemg4.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
41	12, 21, 40, 28	hlexch2 39643	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑉 ∈ (Base‘𝐾)) ∧ ¬ 𝑄 ≤ 𝑉) → (𝑄 ≤ (𝑟 ∨ 𝑉) → 𝑟 ≤ (𝑄 ∨ 𝑉)))
42	7, 9, 10, 18, 39, 41	syl131anc 1385	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ≤ (𝑟 ∨ 𝑉) → 𝑟 ≤ (𝑄 ∨ 𝑉)))
43		simpl32 1256	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑄 ≤ (𝑃 ∨ 𝑉))
44		simp21l 1291	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → 𝑃 ∈ 𝐴)
45	44	adantr 480	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑃 ∈ 𝐴)
46	12, 28	atbase 39549	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
47	45, 46	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑃 ∈ (Base‘𝐾))
48	12, 21, 40	latlej2 18372	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 ≤ (𝑃 ∨ 𝑉))
49	27, 47, 18, 48	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑉 ≤ (𝑃 ∨ 𝑉))
50	12, 40	latjcl 18362	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
51	27, 47, 18, 50	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
52	12, 21, 40	latjle12 18373	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑃 ∨ 𝑉) ∧ 𝑉 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)))
53	27, 31, 18, 51, 52	syl13anc 1374	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ((𝑄 ≤ (𝑃 ∨ 𝑉) ∧ 𝑉 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)))
54	43, 49, 53	mpbi2and 712	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉))
55	12, 28	atbase 39549	. . . . . . . 8 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾))
56	10, 55	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → 𝑟 ∈ (Base‘𝐾))
57	12, 40	latjcl 18362	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑉) ∈ (Base‘𝐾))
58	27, 31, 18, 57	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ∨ 𝑉) ∈ (Base‘𝐾))
59	12, 21	lattr 18367	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑉) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑟 ≤ (𝑄 ∨ 𝑉) ∧ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)) → 𝑟 ≤ (𝑃 ∨ 𝑉)))
60	27, 56, 58, 51, 59	syl13anc 1374	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ((𝑟 ≤ (𝑄 ∨ 𝑉) ∧ (𝑄 ∨ 𝑉) ≤ (𝑃 ∨ 𝑉)) → 𝑟 ≤ (𝑃 ∨ 𝑉)))
61	54, 60	mpan2d 694	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑟 ≤ (𝑄 ∨ 𝑉) → 𝑟 ≤ (𝑃 ∨ 𝑉)))
62	42, 61	syld 47	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑄 ≤ (𝑟 ∨ 𝑉) → 𝑟 ≤ (𝑃 ∨ 𝑉)))
63	6, 62	mtod 198	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → ¬ 𝑄 ≤ (𝑟 ∨ 𝑉))
64		simpl21 1252	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
65		simpl33 1257	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝐹‘(𝐺‘𝑃)) = 𝑃)
66	21, 28, 13, 14, 15, 40, 11	cdlemg6a 40878	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (𝐹‘(𝐺‘𝑟)) = 𝑟)
67	1, 64, 2, 4, 5, 6, 65, 66	syl133anc 1395	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝐹‘(𝐺‘𝑟)) = 𝑟)
68	21, 28, 13, 14, 15, 40, 11	cdlemg6b 40879	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ (𝑟 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑟)) = 𝑟)) → (𝐹‘(𝐺‘𝑄)) = 𝑄)
69	1, 2, 3, 4, 5, 63, 67, 68	syl133anc 1395	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉))) → (𝐹‘(𝐺‘𝑄)) = 𝑄)
70	69	ex 412	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≤ (𝑃 ∨ 𝑉) ∧ (𝐹‘(𝐺‘𝑃)) = 𝑃)) → (((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑉)) → (𝐹‘(𝐺‘𝑄)) = 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  Latclat 18354  Atomscatm 39523  HLchlt 39610  LHypclh 40244  LTrncltrn 40361  trLctrl 40418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-riotaBAD 39213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-undef 8215  df-map 8765  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-p1 18347  df-lat 18355  df-clat 18422  df-oposet 39436  df-ol 39438  df-oml 39439  df-covers 39526  df-ats 39527  df-atl 39558  df-cvlat 39582  df-hlat 39611  df-llines 39758  df-lplanes 39759  df-lvols 39760  df-lines 39761  df-psubsp 39763  df-pmap 39764  df-padd 40056  df-lhyp 40248  df-laut 40249  df-ldil 40364  df-ltrn 40365  df-trl 40419

This theorem is referenced by:  cdlemg6d  40881