cdlemk10 - Mathbox for Norm Megill

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Theorem cdlemk10 41342

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 1142	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simp22 1214	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
4		simp21 1213	. . . . . . . 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 40645	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
8	2, 4, 7	syl2anc 590	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ^◡𝐹 ∈ 𝑇)
9	5, 6	ltrnco 41218	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
10	2, 3, 8, 9	syl3anc 1379	. . . . . 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 40683	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
14	2, 10, 13	syl2anc 590	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
15		simp23 1215	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋 ∈ 𝑇)
16	5, 6	ltrnco 41218	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝑋 ∘ ^◡𝐹) ∈ 𝑇)
17	2, 15, 8, 16	syl3anc 1379	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋 ∘ ^◡𝐹) ∈ 𝑇)
18	11, 5, 6, 12	trlle 40683	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝑋 ∘ ^◡𝐹)) ≤ 𝑊)
19	2, 17, 18	syl2anc 590	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑋 ∘ ^◡𝐹)) ≤ 𝑊)
20		simp1l 1204	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
21	20	hllatd 39863	. . . . . 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 40663	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
24	2, 10, 23	syl2anc 590	. . . . . 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 40663	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝑋 ∘ ^◡𝐹)) ∈ 𝐵)
26	2, 17, 25	syl2anc 590	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑋 ∘ ^◡𝐹)) ∈ 𝐵)
27		simp1r 1205	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
28	22, 5	lhpbase 40497	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
29	27, 28	syl 17	. . . . . 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 18414	. . . . . 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 1380	. . . . 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 718	. . . 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 18403	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵 ∧ (𝑅‘(𝑋 ∘ ^◡𝐹)) ∈ 𝐵) → ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ∈ 𝐵)
35	21, 24, 26, 34	syl3anc 1379	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) ∈ 𝐵)
36		simp3l 1208	. . . . . . 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 40639	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
39	2, 3, 36, 38	syl3anc 1379	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝑃) ∈ 𝐴)
40	11, 37, 5, 6	ltrnat 40639	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑋‘𝑃) ∈ 𝐴)
41	2, 15, 36, 40	syl3anc 1379	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋‘𝑃) ∈ 𝐴)
42	22, 30, 37	hlatjcl 39866	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝐺‘𝑃) ∈ 𝐴 ∧ (𝑋‘𝑃) ∈ 𝐴) → ((𝐺‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵)
43	20, 39, 41, 42	syl3anc 1379	. . . . 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 18434	. . . . 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 1380	. . . 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 15	. . 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 1144	. . . 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 41338	. . . 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 1389	. . 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 5105	. 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 5114	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 class class class wbr 5079 ^◡ccnv 5624 ∘ ccom 5629 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 lecple 17225 joincjn 18275 meetcmee 18276 Latclat 18395 Atomscatm 39762 HLchlt 39849 LHypclh 40483 LTrncltrn 40600 trLctrl 40657

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-riotaBAD 39452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-undef 8220 df-map 8772 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658

This theorem is referenced by: cdlemk11 41348 cdlemk11u 41370

W3C validator