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Theorem 2lnat 34547
Description: Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
Hypotheses
Ref Expression
2lnat.b 𝐵 = (Base‘𝐾)
2lnat.m = (meet‘𝐾)
2lnat.z 0 = (0.‘𝐾)
2lnat.a 𝐴 = (Atoms‘𝐾)
2lnat.n 𝑁 = (Lines‘𝐾)
2lnat.f 𝐹 = (pmap‘𝐾)
Assertion
Ref Expression
2lnat (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)

Proof of Theorem 2lnat
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simp11 1089 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ HL)
2 hlatl 34124 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
31, 2syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ AtLat)
4 hllat 34127 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
51, 4syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ Lat)
6 simp12 1090 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝑋𝐵)
7 simp13 1091 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝑌𝐵)
8 2lnat.b . . . . . 6 𝐵 = (Base‘𝐾)
9 2lnat.m . . . . . 6 = (meet‘𝐾)
108, 9latmcl 16973 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
115, 6, 7, 10syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐵)
12 simp3r 1088 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ≠ 0 )
13 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
14 2lnat.z . . . . 5 0 = (0.‘𝐾)
15 2lnat.a . . . . 5 𝐴 = (Atoms‘𝐾)
168, 13, 14, 15atlex 34080 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑌) ≠ 0 ) → ∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌))
173, 11, 12, 16syl3anc 1323 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → ∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌))
18 simp13l 1174 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑋𝑌)
19 simp11 1089 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵))
20 simp12l 1172 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐹𝑋) ∈ 𝑁)
21 simp12r 1173 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐹𝑌) ∈ 𝑁)
22 2lnat.n . . . . . . . . . . 11 𝑁 = (Lines‘𝐾)
23 2lnat.f . . . . . . . . . . 11 𝐹 = (pmap‘𝐾)
248, 13, 22, 23lncmp 34546 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁)) → (𝑋(le‘𝐾)𝑌𝑋 = 𝑌))
2519, 20, 21, 24syl12anc 1321 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋(le‘𝐾)𝑌𝑋 = 𝑌))
26 simp111 1188 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ HL)
2726, 4syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ Lat)
28 simp112 1189 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑋𝐵)
29 simp113 1190 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑌𝐵)
308, 13, 9latleeqm1 17000 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 𝑌) = 𝑋))
3127, 28, 29, 30syl3anc 1323 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 𝑌) = 𝑋))
3225, 31bitr3d 270 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 = 𝑌 ↔ (𝑋 𝑌) = 𝑋))
3332necon3bid 2834 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋𝑌 ↔ (𝑋 𝑌) ≠ 𝑋))
3418, 33mpbid 222 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌) ≠ 𝑋)
35 simp3 1061 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌))
368, 13, 9latmle1 16997 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
3727, 28, 29, 36syl3anc 1323 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌)(le‘𝐾)𝑋)
38 hlpos 34129 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3926, 38syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ Poset)
408, 15atbase 34053 . . . . . . . . . . 11 (𝑝𝐴𝑝𝐵)
41403ad2ant2 1081 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝𝐵)
4227, 28, 29, 10syl3anc 1323 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌) ∈ 𝐵)
43 simp2 1060 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝𝐴)
448, 13, 27, 41, 42, 28, 35, 37lattrd 16979 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)𝑋)
45 eqid 2621 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
468, 13, 45, 15, 22, 23lncvrat 34545 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) ∧ ((𝐹𝑋) ∈ 𝑁𝑝(le‘𝐾)𝑋)) → 𝑝( ⋖ ‘𝐾)𝑋)
4726, 28, 43, 20, 44, 46syl32anc 1331 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
488, 13, 45cvrnbtwn4 34043 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑝𝐵𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) ∧ 𝑝( ⋖ ‘𝐾)𝑋) → ((𝑝(le‘𝐾)(𝑋 𝑌) ∧ (𝑋 𝑌)(le‘𝐾)𝑋) ↔ (𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋)))
4939, 41, 28, 42, 47, 48syl131anc 1336 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → ((𝑝(le‘𝐾)(𝑋 𝑌) ∧ (𝑋 𝑌)(le‘𝐾)𝑋) ↔ (𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋)))
5035, 37, 49mpbi2and 955 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋))
51 neor 2881 . . . . . . . 8 ((𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋) ↔ (𝑝 ≠ (𝑋 𝑌) → (𝑋 𝑌) = 𝑋))
5250, 51sylib 208 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑝 ≠ (𝑋 𝑌) → (𝑋 𝑌) = 𝑋))
5352necon1d 2812 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → ((𝑋 𝑌) ≠ 𝑋𝑝 = (𝑋 𝑌)))
5434, 53mpd 15 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝 = (𝑋 𝑌))
55543exp 1261 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑝𝐴 → (𝑝(le‘𝐾)(𝑋 𝑌) → 𝑝 = (𝑋 𝑌))))
5655reximdvai 3009 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌) → ∃𝑝𝐴 𝑝 = (𝑋 𝑌)))
5717, 56mpd 15 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → ∃𝑝𝐴 𝑝 = (𝑋 𝑌))
58 risset 3055 . 2 ((𝑋 𝑌) ∈ 𝐴 ↔ ∃𝑝𝐴 𝑝 = (𝑋 𝑌))
5957, 58sylibr 224 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  Posetcpo 16861  meetcmee 16866  0.cp0 16958  Latclat 16966  ccvr 34026  Atomscatm 34027  AtLatcal 34028  HLchlt 34114  Linesclines 34257  pmapcpmap 34260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-lines 34264  df-pmap 34267
This theorem is referenced by:  cdleme3h  34999  cdleme7ga  35012
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