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Theorem llncmp 34285
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 1060 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋𝑁)
2 simp1 1059 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝐾 ∈ HL)
3 eqid 2621 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
4 llncmp.n . . . . . . 7 𝑁 = (LLines‘𝐾)
53, 4llnbase 34272 . . . . . 6 (𝑋𝑁𝑋 ∈ (Base‘𝐾))
653ad2ant2 1081 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋 ∈ (Base‘𝐾))
7 eqid 2621 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8 eqid 2621 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
93, 7, 8, 4islln4 34270 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
102, 6, 9syl2anc 692 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
111, 10mpbid 222 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋)
12 simpr3 1067 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 𝑌)
13 hlpos 34129 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Poset)
14133ad2ant1 1080 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝐾 ∈ Poset)
1514adantr 481 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝐾 ∈ Poset)
166adantr 481 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17 simpl3 1064 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑌𝑁)
183, 4llnbase 34272 . . . . . . . 8 (𝑌𝑁𝑌 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑌 ∈ (Base‘𝐾))
20 simpr1 1065 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
213, 8atbase 34053 . . . . . . . 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 1066 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
24 simpl1 1062 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝐾 ∈ HL)
25 llncmp.l . . . . . . . . . . 11 = (le‘𝐾)
263, 25, 7cvrle 34042 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑝( ⋖ ‘𝐾)𝑋) → 𝑝 𝑋)
2724, 22, 16, 23, 26syl31anc 1326 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 𝑋)
283, 25postr 16874 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
2915, 22, 16, 19, 28syl13anc 1325 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
3027, 12, 29mp2and 714 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 𝑌)
3125, 7, 8, 4atcvrlln2 34282 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑌𝑁) ∧ 𝑝 𝑌) → 𝑝( ⋖ ‘𝐾)𝑌)
3224, 20, 17, 30, 31syl31anc 1326 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑌)
333, 25, 7cvrcmp 34047 . . . . . . 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 1341 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
3512, 34mpbid 222 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 = 𝑌)
36353exp2 1282 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝( ⋖ ‘𝐾)𝑋 → (𝑋 𝑌𝑋 = 𝑌))))
3736rexlimdv 3023 . . 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 16872 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 𝑋)
4014, 6, 39syl2anc 692 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋 𝑋)
41 breq2 4617 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
4240, 41syl5ibcom 235 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 = 𝑌𝑋 𝑌))
4338, 42impbid 202 1 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  Posetcpo 16861  ccvr 34026  Atomscatm 34027  HLchlt 34114  LLinesclln 34254
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-llines 34261
This theorem is referenced by:  llnnlt  34286  2llnmat  34287  llnmlplnN  34302  dalem16  34442  dalem60  34495  llnexchb2  34632
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