cdlemk10 - Mathbox for Norm Megill

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Theorem cdlemk10 42092

Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-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.v1	⊢ 𝑉 = (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))))

**Assertion**
Ref	Expression
cdlemk10	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑉 ≤ (𝑅‘(𝑋 ∘ ^◡𝐺)))

**Proof of Theorem cdlemk10**
Step	Hyp	Ref	Expression
1		cdlemk.v1	. 2 ⊢ 𝑉 = (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))))
2		simp1 1154	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simp22 1226	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
4		simp21 1225	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
5		cdlemk.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdlemk.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7	5, 6	ltrncnv 41395	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
8	2, 4, 7	syl2anc 596	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ^◡𝐹 ∈ 𝑇)
9	5, 6	ltrnco 41968	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
10	2, 3, 8, 9	syl3anc 1399	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
11		cdlemk.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
12		cdlemk.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
13	11, 5, 6, 12	trlle 41433	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
14	2, 10, 13	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
15		simp23 1227	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋 ∈ 𝑇)
16	5, 6	ltrnco 41968	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝑋 ∘ ^◡𝐹) ∈ 𝑇)
17	2, 15, 8, 16	syl3anc 1399	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋 ∘ ^◡𝐹) ∈ 𝑇)
18	11, 5, 6, 12	trlle 41433	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝑋 ∘ ^◡𝐹)) ≤ 𝑊)
19	2, 17, 18	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑋 ∘ ^◡𝐹)) ≤ 𝑊)
20		simp1l 1216	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
21	20	hllatd 40613	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat)
22		cdlemk.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
23	22, 5, 6, 12	trlcl 41413	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
24	2, 10, 23	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
25	22, 5, 6, 12	trlcl 41413	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝑋 ∘ ^◡𝐹)) ∈ 𝐵)
26	2, 17, 25	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑋 ∘ ^◡𝐹)) ∈ 𝐵)
27		simp1r 1217	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
28	22, 5	lhpbase 41247	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
29	27, 28	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
30		cdlemk.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
31	22, 11, 30	latjle12 18653	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵 ∧ (𝑅‘(𝑋 ∘ ^◡𝐹)) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (((𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊 ∧ (𝑅‘(𝑋 ∘ ^◡𝐹)) ≤ 𝑊) ↔ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ≤ 𝑊))
32	21, 24, 26, 29, 31	syl13anc 1400	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊 ∧ (𝑅‘(𝑋 ∘ ^◡𝐹)) ≤ 𝑊) ↔ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ≤ 𝑊))
33	14, 19, 32	mpbi2and 725	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ≤ 𝑊)
34	22, 30	latjcl 18642	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵 ∧ (𝑅‘(𝑋 ∘ ^◡𝐹)) ∈ 𝐵) → ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ∈ 𝐵)
35	21, 24, 26, 34	syl3anc 1399	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ∈ 𝐵)
36		simp3l 1220	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
37		cdlemk.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
38	11, 37, 5, 6	ltrnat 41389	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
39	2, 3, 36, 38	syl3anc 1399	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝑃) ∈ 𝐴)
40	11, 37, 5, 6	ltrnat 41389	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑋‘𝑃) ∈ 𝐴)
41	2, 15, 36, 40	syl3anc 1399	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋‘𝑃) ∈ 𝐴)
42	22, 30, 37	hlatjcl 40616	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝐺‘𝑃) ∈ 𝐴 ∧ (𝑋‘𝑃) ∈ 𝐴) → ((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵)
43	20, 39, 41, 42	syl3anc 1399	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵)
44		cdlemk.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
45	22, 11, 44	latmlem2 18673	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ ((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵)) → (((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ≤ 𝑊 → (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹)))) ≤ (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊)))
46	21, 35, 29, 43, 45	syl13anc 1400	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ≤ 𝑊 → (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹)))) ≤ (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊)))
47	33, 46	mpd 16	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹)))) ≤ (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊))
48		simp3 1156	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
49	22, 11, 30, 37, 5, 6, 12, 44	cdlemk9 42088	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊) = (𝑅‘(𝑋 ∘ ^◡𝐺)))
50	20, 27, 3, 15, 48, 49	syl221anc 1409	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊) = (𝑅‘(𝑋 ∘ ^◡𝐺)))
51	47, 50	breqtrd 5134	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹)))) ≤ (𝑅‘(𝑋 ∘ ^◡𝐺)))
52	1, 51	eqbrtrid 5143	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑉 ≤ (𝑅‘(𝑋 ∘ ^◡𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 class class class wbr 5106 ^◡ccnv 5654 ∘ ccom 5659 ‘cfv 6558 (class class class)co 7443 Basecbs 17416 lecple 17464 joincjn 18514 meetcmee 18515 Latclat 18634 Atomscatm 40512 HLchlt 40599 LHypclh 41233 LTrncltrn 41350 trLctrl 41407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-riotaBAD 40202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-1st 8025 df-2nd 8026 df-undef 8318 df-map 8877 df-proset 18497 df-poset 18516 df-plt 18531 df-lub 18547 df-glb 18548 df-join 18549 df-meet 18550 df-p0 18626 df-p1 18627 df-lat 18635 df-clat 18702 df-oposet 40425 df-ol 40427 df-oml 40428 df-covers 40515 df-ats 40516 df-atl 40547 df-cvlat 40571 df-hlat 40600 df-llines 40747 df-lplanes 40748 df-lvols 40749 df-lines 40750 df-psubsp 40752 df-pmap 40753 df-padd 41045 df-lhyp 41237 df-laut 41238 df-ldil 41353 df-ltrn 41354 df-trl 41408

This theorem is used by: cdlemk11 42098 cdlemk11u 42120

W3C validator