Theorem cdlemg6c 38277

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 771	. . 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 773	. . . 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 484	. . . . . 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 779	. . . . . 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 2738	. . . . . . . . 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 37821	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
17	1, 5, 16	syl2anc 587	. . . . . . 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 2837	. . . . . 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 484	. . . . . . 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 37841	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ≤ 𝑊)
23	1, 5, 22	syl2anc 587	. . . . . . . . 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 5065	. . . . . . . 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 37021	. . . . . . . . . 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 484	. . . . . . . . 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 36946	. . . . . . . . . . 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 484	. . . . . . . . 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 37655	. . . . . . . . . . 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 484	. . . . . . . . 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 17782	. . . . . . . . 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 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))) → ((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 201	. . . . . 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 37040	. . . . . 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 1384	. . . . 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 484	. . . . . . . . 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 36946	. . . . . . . . 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 17787	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → 𝑉 ≤ (𝑃 ∨ 𝑉))
49	27, 47, 18, 48	syl3anc 1372	. . . . . . 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 17777	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
51	27, 47, 18, 50	syl3anc 1372	. . . . . . . 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 17788	. . . . . . . 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 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 ≤ (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 36946	. . . . . . . 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 17777	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑉) ∈ (Base‘𝐾))
58	27, 31, 18, 57	syl3anc 1372	. . . . . . 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 17782	. . . . . . 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 1373	. . . . . 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 201	. . 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 38275	. . . 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 1394	. . 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 38276	. . 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 1394	. 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 416	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 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   class class class wbr 5030  ‘cfv 6339  (class class class)co 7170  Basecbs 16586  lecple 16675  joincjn 17670  Latclat 17771  Atomscatm 36920  HLchlt 37007  LHypclh 37641  LTrncltrn 37758  trLctrl 37815

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-riotaBAD 36610

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-undef 7968  df-map 8439  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-p1 17766  df-lat 17772  df-clat 17834  df-oposet 36833  df-ol 36835  df-oml 36836  df-covers 36923  df-ats 36924  df-atl 36955  df-cvlat 36979  df-hlat 37008  df-llines 37155  df-lplanes 37156  df-lvols 37157  df-lines 37158  df-psubsp 37160  df-pmap 37161  df-padd 37453  df-lhyp 37645  df-laut 37646  df-ldil 37761  df-ltrn 37762  df-trl 37816

This theorem is referenced by:  cdlemg6d  38278