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Theorem cdleme35f 38124
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
Hypotheses
Ref Expression
cdleme35.l = (le‘𝐾)
cdleme35.j = (join‘𝐾)
cdleme35.m = (meet‘𝐾)
cdleme35.a 𝐴 = (Atoms‘𝐾)
cdleme35.h 𝐻 = (LHyp‘𝐾)
cdleme35.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme35.f 𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
Assertion
Ref Expression
cdleme35f ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑅 𝑈) (𝑃 𝑅)) = 𝑅)

Proof of Theorem cdleme35f
StepHypRef Expression
1 simp11l 1285 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝐾 ∈ HL)
2 simp12l 1287 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃𝐴)
3 simp2rl 1243 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅𝐴)
4 cdleme35.j . . . . 5 = (join‘𝐾)
5 cdleme35.a . . . . 5 𝐴 = (Atoms‘𝐾)
64, 5hlatjcom 37038 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) = (𝑅 𝑃))
71, 2, 3, 6syl3anc 1372 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃 𝑅) = (𝑅 𝑃))
87oveq2d 7199 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑅 𝑈) (𝑃 𝑅)) = ((𝑅 𝑈) (𝑅 𝑃)))
9 simp11 1204 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
10 simp12 1205 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
11 simp13l 1289 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑄𝐴)
12 simp2l 1200 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃𝑄)
13 cdleme35.l . . . . 5 = (le‘𝐾)
14 cdleme35.m . . . . 5 = (meet‘𝐾)
15 cdleme35.h . . . . 5 𝐻 = (LHyp‘𝐾)
16 cdleme35.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
1713, 4, 14, 5, 15, 16cdleme0a 37881 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
189, 10, 11, 12, 17syl112anc 1375 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑈𝐴)
19 simp12r 1288 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑃 𝑊)
201hllatd 37034 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝐾 ∈ Lat)
21 eqid 2739 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
2221, 4, 5hlatjcl 37037 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
231, 2, 11, 22syl3anc 1372 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃 𝑄) ∈ (Base‘𝐾))
24 simp11r 1286 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑊𝐻)
2521, 15lhpbase 37668 . . . . . . . . 9 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2624, 25syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑊 ∈ (Base‘𝐾))
2721, 13, 14latmle2 17816 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
2820, 23, 26, 27syl3anc 1372 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑄) 𝑊) 𝑊)
2916, 28eqbrtrid 5075 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑈 𝑊)
30 breq1 5043 . . . . . 6 (𝑈 = 𝑃 → (𝑈 𝑊𝑃 𝑊))
3129, 30syl5ibcom 248 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑈 = 𝑃𝑃 𝑊))
3231necon3bd 2949 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (¬ 𝑃 𝑊𝑈𝑃))
3319, 32mpd 15 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑈𝑃)
34 simp3 1139 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑅 (𝑃 𝑄))
3521, 13, 14latmle1 17815 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
3620, 23, 26, 35syl3anc 1372 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
3716, 36eqbrtrid 5075 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑈 (𝑃 𝑄))
3813, 4, 5hlatlej1 37045 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
391, 2, 11, 38syl3anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃 (𝑃 𝑄))
4021, 5atbase 36959 . . . . . . . 8 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
4118, 40syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑈 ∈ (Base‘𝐾))
4221, 5atbase 36959 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
432, 42syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃 ∈ (Base‘𝐾))
4421, 13, 4latjle12 17801 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑈 (𝑃 𝑄) ∧ 𝑃 (𝑃 𝑄)) ↔ (𝑈 𝑃) (𝑃 𝑄)))
4520, 41, 43, 23, 44syl13anc 1373 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑈 (𝑃 𝑄) ∧ 𝑃 (𝑃 𝑄)) ↔ (𝑈 𝑃) (𝑃 𝑄)))
4637, 39, 45mpbi2and 712 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑈 𝑃) (𝑃 𝑄))
4721, 5atbase 36959 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
483, 47syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 ∈ (Base‘𝐾))
4921, 4, 5hlatjcl 37037 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑈𝐴𝑃𝐴) → (𝑈 𝑃) ∈ (Base‘𝐾))
501, 18, 2, 49syl3anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑈 𝑃) ∈ (Base‘𝐾))
5121, 13lattr 17795 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝑈 𝑃) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑅 (𝑈 𝑃) ∧ (𝑈 𝑃) (𝑃 𝑄)) → 𝑅 (𝑃 𝑄)))
5220, 48, 50, 23, 51syl13anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑅 (𝑈 𝑃) ∧ (𝑈 𝑃) (𝑃 𝑄)) → 𝑅 (𝑃 𝑄)))
5346, 52mpan2d 694 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 (𝑈 𝑃) → 𝑅 (𝑃 𝑄)))
5434, 53mtod 201 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑅 (𝑈 𝑃))
5513, 4, 14, 52llnma2 37459 . . 3 ((𝐾 ∈ HL ∧ (𝑈𝐴𝑃𝐴𝑅𝐴) ∧ (𝑈𝑃 ∧ ¬ 𝑅 (𝑈 𝑃))) → ((𝑅 𝑈) (𝑅 𝑃)) = 𝑅)
561, 18, 2, 3, 33, 54, 55syl132anc 1389 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑅 𝑈) (𝑅 𝑃)) = 𝑅)
578, 56eqtrd 2774 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑅 𝑈) (𝑃 𝑅)) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2935   class class class wbr 5040  cfv 6350  (class class class)co 7183  Basecbs 16599  lecple 16688  joincjn 17683  meetcmee 17684  Latclat 17784  Atomscatm 36933  HLchlt 37020  LHypclh 37654
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 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7140  df-ov 7186  df-oprab 7187  df-proset 17667  df-poset 17685  df-plt 17697  df-lub 17713  df-glb 17714  df-join 17715  df-meet 17716  df-p0 17778  df-p1 17779  df-lat 17785  df-clat 17847  df-oposet 36846  df-ol 36848  df-oml 36849  df-covers 36936  df-ats 36937  df-atl 36968  df-cvlat 36992  df-hlat 37021  df-lhyp 37658
This theorem is referenced by:  cdleme35g  38125
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