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Theorem llncmp 35478
Description: If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)
Hypotheses
Ref Expression
llncmp.l = (le‘𝐾)
llncmp.n 𝑁 = (LLines‘𝐾)
Assertion
Ref Expression
llncmp ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 𝑌𝑋 = 𝑌))

Proof of Theorem llncmp
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simp2 1167 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋𝑁)
2 simp1 1166 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝐾 ∈ HL)
3 eqid 2765 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
4 llncmp.n . . . . . . 7 𝑁 = (LLines‘𝐾)
53, 4llnbase 35465 . . . . . 6 (𝑋𝑁𝑋 ∈ (Base‘𝐾))
653ad2ant2 1164 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋 ∈ (Base‘𝐾))
7 eqid 2765 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8 eqid 2765 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
93, 7, 8, 4islln4 35463 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
102, 6, 9syl2anc 579 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
111, 10mpbid 223 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋)
12 simpr3 1252 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 𝑌)
13 hlpos 35322 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Poset)
14133ad2ant1 1163 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝐾 ∈ Poset)
1514adantr 472 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝐾 ∈ Poset)
166adantr 472 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17 simpl3 1246 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑌𝑁)
183, 4llnbase 35465 . . . . . . . 8 (𝑌𝑁𝑌 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑌 ∈ (Base‘𝐾))
20 simpr1 1248 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
213, 8atbase 35245 . . . . . . . 8 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
2220, 21syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 ∈ (Base‘𝐾))
23 simpr2 1250 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
24 simpl1 1242 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝐾 ∈ HL)
25 llncmp.l . . . . . . . . . . 11 = (le‘𝐾)
263, 25, 7cvrle 35234 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑝( ⋖ ‘𝐾)𝑋) → 𝑝 𝑋)
2724, 22, 16, 23, 26syl31anc 1492 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 𝑋)
283, 25postr 17219 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
2915, 22, 16, 19, 28syl13anc 1491 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
3027, 12, 29mp2and 690 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 𝑌)
3125, 7, 8, 4atcvrlln2 35475 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑌𝑁) ∧ 𝑝 𝑌) → 𝑝( ⋖ ‘𝐾)𝑌)
3224, 20, 17, 30, 31syl31anc 1492 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑌)
333, 25, 7cvrcmp 35239 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) ∧ (𝑝( ⋖ ‘𝐾)𝑋𝑝( ⋖ ‘𝐾)𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
3415, 16, 19, 22, 23, 32, 33syl132anc 1507 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
3512, 34mpbid 223 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 = 𝑌)
36353exp2 1463 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝( ⋖ ‘𝐾)𝑋 → (𝑋 𝑌𝑋 = 𝑌))))
3736rexlimdv 3177 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋 → (𝑋 𝑌𝑋 = 𝑌)))
3811, 37mpd 15 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 𝑌𝑋 = 𝑌))
393, 25posref 17217 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 𝑋)
4014, 6, 39syl2anc 579 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋 𝑋)
41 breq2 4813 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
4240, 41syl5ibcom 236 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 = 𝑌𝑋 𝑌))
4338, 42impbid 203 1 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wrex 3056   class class class wbr 4809  cfv 6068  Basecbs 16130  lecple 16221  Posetcpo 17206  ccvr 35218  Atomscatm 35219  HLchlt 35306  LLinesclln 35447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-proset 17194  df-poset 17212  df-plt 17224  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-p0 17305  df-lat 17312  df-clat 17374  df-oposet 35132  df-ol 35134  df-oml 35135  df-covers 35222  df-ats 35223  df-atl 35254  df-cvlat 35278  df-hlat 35307  df-llines 35454
This theorem is referenced by:  llnnlt  35479  2llnmat  35480  llnmlplnN  35495  dalem16  35635  dalem60  35688  llnexchb2  35825
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