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Theorem cdleme19d 35911
Description: Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114. 𝐷, 𝐹, 𝐺 represent s2, f(s), f(t). We prove f(s) s2 = f(s) f(t). (Contributed by NM, 14-Nov-2012.)
Hypotheses
Ref Expression
cdleme19.l = (le‘𝐾)
cdleme19.j = (join‘𝐾)
cdleme19.m = (meet‘𝐾)
cdleme19.a 𝐴 = (Atoms‘𝐾)
cdleme19.h 𝐻 = (LHyp‘𝐾)
cdleme19.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme19.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme19.g 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
cdleme19.d 𝐷 = ((𝑅 𝑆) 𝑊)
cdleme19.y 𝑌 = ((𝑅 𝑇) 𝑊)
Assertion
Ref Expression
cdleme19d ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝐹 𝐷) = (𝐹 𝐺))

Proof of Theorem cdleme19d
StepHypRef Expression
1 cdleme19.l . . 3 = (le‘𝐾)
2 cdleme19.j . . 3 = (join‘𝐾)
3 cdleme19.m . . 3 = (meet‘𝐾)
4 cdleme19.a . . 3 𝐴 = (Atoms‘𝐾)
5 cdleme19.h . . 3 𝐻 = (LHyp‘𝐾)
6 cdleme19.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
7 cdleme19.f . . 3 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
8 cdleme19.g . . 3 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
9 cdleme19.d . . 3 𝐷 = ((𝑅 𝑆) 𝑊)
10 cdleme19.y . . 3 𝑌 = ((𝑅 𝑇) 𝑊)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme19b 35909 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐷 (𝐹 𝐺))
12 simp11l 1192 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐾 ∈ HL)
13 hlcvl 34964 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
1412, 13syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐾 ∈ CvLat)
15 simp11r 1193 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑊𝐻)
16 simp21l 1198 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑆𝐴)
17 simp21r 1199 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → ¬ 𝑆 𝑊)
18 simp23 1116 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑅𝐴)
19 simp33l 1208 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑅 (𝑃 𝑄))
20 simp32l 1206 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → ¬ 𝑆 (𝑃 𝑄))
211, 2, 3, 4, 5, 9cdlemeda 35903 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐷𝐴)
2212, 15, 16, 17, 18, 19, 20, 21syl223anc 1392 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐷𝐴)
23 simp11 1111 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
24 simp12 1112 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
25 simp13 1113 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
26 simp22 1115 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
27 simp31l 1204 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑃𝑄)
28 simp32r 1207 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → ¬ 𝑇 (𝑃 𝑄))
291, 2, 3, 4, 5, 6, 8cdleme3fa 35841 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑇 (𝑃 𝑄))) → 𝐺𝐴)
3023, 24, 25, 26, 27, 28, 29syl132anc 1384 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐺𝐴)
31 simp21 1114 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
321, 2, 3, 4, 5, 6, 7cdleme3fa 35841 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐹𝐴)
3323, 24, 25, 31, 27, 20, 32syl132anc 1384 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐹𝐴)
341, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme19c 35910 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐹𝐷)
3512, 15, 24, 25, 31, 18, 27, 20, 34syl233anc 1395 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐹𝐷)
3635necomd 2878 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐷𝐹)
371, 2, 4cvlatexchb1 34939 . . 3 ((𝐾 ∈ CvLat ∧ (𝐷𝐴𝐺𝐴𝐹𝐴) ∧ 𝐷𝐹) → (𝐷 (𝐹 𝐺) ↔ (𝐹 𝐷) = (𝐹 𝐺)))
3814, 22, 30, 33, 36, 37syl131anc 1379 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝐷 (𝐹 𝐺) ↔ (𝐹 𝐷) = (𝐹 𝐺)))
3911, 38mpbid 222 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝐹 𝐷) = (𝐹 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  CvLatclc 34870  HLchlt 34955  LHypclh 35588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592
This theorem is referenced by:  cdleme19e  35912
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