cdlemki - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemki		Structured version Visualization version GIF version

Theorem cdlemki 40346

Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: Eliminate and put into cdlemksel 40350. (Contributed by NM, 25-Jun-2013.)

**Hypotheses**
Ref	Expression
cdlemk.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk.l	⊢ ≤ = (le‘𝐾)
cdlemk.j	⊢ ∨ = (join‘𝐾)
cdlemk.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk.m	⊢ ∧ = (meet‘𝐾)
cdlemk.i	⊢ 𝐼 = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))

**Assertion**
Ref	Expression
cdlemki	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → 𝐼 ∈ 𝑇)

Distinct variable groups: ∧ ,𝑖 ≤ ,𝑖 ∨ ,𝑖 𝐴,𝑖 𝑖,𝐹 𝑖,𝐻 𝑖,𝐾 𝑖,𝑁 𝑃,𝑖 𝑅,𝑖 𝑇,𝑖 𝑖,𝑊 𝑖,𝐺 Allowed substitution hints: 𝐵(𝑖) 𝐼(𝑖)

**Proof of Theorem cdlemki**
Step	Hyp	Ref	Expression
1		simp11 1200	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp22 1204	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simp1 1133	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇))
4		simp21 1203	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → 𝑁 ∈ 𝑇)
5		cdlemk.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
6		cdlemk.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemk.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemk.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9	5, 6, 7, 8	ltrnel 39644	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑁‘𝑃) ∈ 𝐴 ∧ ¬ (𝑁‘𝑃) ≤ 𝑊))
10	1, 4, 2, 9	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → ((𝑁‘𝑃) ∈ 𝐴 ∧ ¬ (𝑁‘𝑃) ≤ 𝑊))
11		simp11l 1281	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → 𝐾 ∈ HL)
12		simp22l 1289	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → 𝑃 ∈ 𝐴)
13	9	simpld 493	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑁‘𝑃) ∈ 𝐴)
14	1, 4, 2, 13	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ∈ 𝐴)
15		cdlemk.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
16	5, 15, 6	hlatlej2 38880	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝑁‘𝑃) ∈ 𝐴) → (𝑁‘𝑃) ≤ (𝑃 ∨ (𝑁‘𝑃)))
17	11, 12, 14, 16	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ≤ (𝑃 ∨ (𝑁‘𝑃)))
18		simp23 1205	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑅‘𝐹) = (𝑅‘𝑁))
19	18	oveq2d 7442	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝑅‘𝑁)))
20		cdlemk.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
21	5, 15, 6, 7, 8, 20	trljat1 39671	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝑁)) = (𝑃 ∨ (𝑁‘𝑃)))
22	1, 4, 2, 21	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑃 ∨ (𝑅‘𝑁)) = (𝑃 ∨ (𝑁‘𝑃)))
23	19, 22	eqtr2d 2769	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑃 ∨ (𝑁‘𝑃)) = (𝑃 ∨ (𝑅‘𝐹)))
24	17, 23	breqtrd 5178	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)))
25		simp31 1206	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → 𝐹 ≠ ( I ↾ 𝐵))
26		simp32 1207	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → 𝐺 ≠ ( I ↾ 𝐵))
27		simp33 1208	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑅‘𝐺) ≠ (𝑅‘𝐹))
28	27	necomd 2993	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (𝑅‘𝐹) ≠ (𝑅‘𝐺))
29		cdlemk.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
30		cdlemk.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
31		eqid 2728	. . . 4 ⊢ ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
32	29, 5, 15, 30, 6, 7, 8, 20, 31	cdlemh 40322	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ ((𝑁‘𝑃) ∈ 𝐴 ∧ ¬ (𝑁‘𝑃) ≤ 𝑊) ∧ (𝑁‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∈ 𝐴 ∧ ¬ ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ≤ 𝑊))
33	3, 2, 10, 24, 25, 26, 28, 32	syl133anc 1390	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∈ 𝐴 ∧ ¬ ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ≤ 𝑊))
34		cdlemk.i	. . 3 ⊢ 𝐼 = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
35	5, 6, 7, 8, 34	ltrniotacl 40084	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∈ 𝐴 ∧ ¬ ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ≤ 𝑊)) → 𝐼 ∈ 𝑇)
36	1, 2, 33, 35	syl3anc 1368	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹))) → 𝐼 ∈ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 I cid 5579 ^◡ccnv 5681 ↾ cres 5684 ∘ ccom 5686 ‘cfv 6553 ℩crio 7381 (class class class)co 7426 Basecbs 17187 lecple 17247 joincjn 18310 meetcmee 18311 Atomscatm 38767 HLchlt 38854 LHypclh 39489 LTrncltrn 39606 trLctrl 39663

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-riotaBAD 38457

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-undef 8285 df-map 8853 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664

This theorem is referenced by: cdlemksel 40350

W3C validator

	
		OSZAR »