Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2lnat Structured version   Visualization version   GIF version

Theorem 2lnat 36806
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 1197 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ HL)
2 hlatl 36382 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
31, 2syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ AtLat)
41hllatd 36386 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝐾 ∈ Lat)
5 simp12 1198 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝑋𝐵)
6 simp13 1199 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → 𝑌𝐵)
7 2lnat.b . . . . . 6 𝐵 = (Base‘𝐾)
8 2lnat.m . . . . . 6 = (meet‘𝐾)
97, 8latmcl 17657 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
104, 5, 6, 9syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐵)
11 simp3r 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ≠ 0 )
12 eqid 2826 . . . . 5 (le‘𝐾) = (le‘𝐾)
13 2lnat.z . . . . 5 0 = (0.‘𝐾)
14 2lnat.a . . . . 5 𝐴 = (Atoms‘𝐾)
157, 12, 13, 14atlex 36338 . . . 4 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑌) ≠ 0 ) → ∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌))
163, 10, 11, 15syl3anc 1365 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → ∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌))
17 simp13l 1282 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑋𝑌)
18 simp11 1197 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵))
19 simp12l 1280 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐹𝑋) ∈ 𝑁)
20 simp12r 1281 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝐹𝑌) ∈ 𝑁)
21 2lnat.n . . . . . . . . . . 11 𝑁 = (Lines‘𝐾)
22 2lnat.f . . . . . . . . . . 11 𝐹 = (pmap‘𝐾)
237, 12, 21, 22lncmp 36805 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁)) → (𝑋(le‘𝐾)𝑌𝑋 = 𝑌))
2418, 19, 20, 23syl12anc 834 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋(le‘𝐾)𝑌𝑋 = 𝑌))
25 simp111 1296 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ HL)
2625hllatd 36386 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ Lat)
27 simp112 1297 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑋𝐵)
28 simp113 1298 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑌𝐵)
297, 12, 8latleeqm1 17684 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 𝑌) = 𝑋))
3026, 27, 28, 29syl3anc 1365 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋(le‘𝐾)𝑌 ↔ (𝑋 𝑌) = 𝑋))
3124, 30bitr3d 282 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 = 𝑌 ↔ (𝑋 𝑌) = 𝑋))
3231necon3bid 3065 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋𝑌 ↔ (𝑋 𝑌) ≠ 𝑋))
3317, 32mpbid 233 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌) ≠ 𝑋)
34 simp3 1132 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌))
357, 12, 8latmle1 17681 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
3626, 27, 28, 35syl3anc 1365 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌)(le‘𝐾)𝑋)
37 hlpos 36388 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Poset)
3825, 37syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝐾 ∈ Poset)
397, 14atbase 36311 . . . . . . . . . . 11 (𝑝𝐴𝑝𝐵)
40393ad2ant2 1128 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝𝐵)
4126, 27, 28, 9syl3anc 1365 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑋 𝑌) ∈ 𝐵)
42 simp2 1131 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝𝐴)
437, 12, 26, 40, 41, 27, 34, 36lattrd 17663 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)𝑋)
44 eqid 2826 . . . . . . . . . . . 12 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
457, 12, 44, 14, 21, 22lncvrat 36804 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑝𝐴) ∧ ((𝐹𝑋) ∈ 𝑁𝑝(le‘𝐾)𝑋)) → 𝑝( ⋖ ‘𝐾)𝑋)
4625, 27, 42, 19, 43, 45syl32anc 1372 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
477, 12, 44cvrnbtwn4 36301 . . . . . . . . . 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)))
4838, 40, 27, 41, 46, 47syl131anc 1377 . . . . . . . . 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)))
4934, 36, 48mpbi2and 708 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋))
50 neor 3113 . . . . . . . 8 ((𝑝 = (𝑋 𝑌) ∨ (𝑋 𝑌) = 𝑋) ↔ (𝑝 ≠ (𝑋 𝑌) → (𝑋 𝑌) = 𝑋))
5149, 50sylib 219 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → (𝑝 ≠ (𝑋 𝑌) → (𝑋 𝑌) = 𝑋))
5251necon1d 3043 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → ((𝑋 𝑌) ≠ 𝑋𝑝 = (𝑋 𝑌)))
5333, 52mpd 15 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) ∧ 𝑝𝐴𝑝(le‘𝐾)(𝑋 𝑌)) → 𝑝 = (𝑋 𝑌))
54533exp 1113 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑝𝐴 → (𝑝(le‘𝐾)(𝑋 𝑌) → 𝑝 = (𝑋 𝑌))))
5554reximdvai 3277 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (∃𝑝𝐴 𝑝(le‘𝐾)(𝑋 𝑌) → ∃𝑝𝐴 𝑝 = (𝑋 𝑌)))
5616, 55mpd 15 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → ∃𝑝𝐴 𝑝 = (𝑋 𝑌))
57 risset 3272 . 2 ((𝑋 𝑌) ∈ 𝐴 ↔ ∃𝑝𝐴 𝑝 = (𝑋 𝑌))
5856, 57sylibr 235 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 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wrex 3144   class class class wbr 5063  cfv 6354  (class class class)co 7150  Basecbs 16478  lecple 16567  Posetcpo 17545  meetcmee 17550  0.cp0 17642  Latclat 17650  ccvr 36284  Atomscatm 36285  AtLatcal 36286  HLchlt 36372  Linesclines 36516  pmapcpmap 36519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-lat 17651  df-clat 17713  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-lines 36523  df-pmap 36526
This theorem is referenced by:  cdleme3h  37257  cdleme7ga  37270
  Copyright terms: Public domain W3C validator