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Theorem llncmp 36650
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 1131 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋𝑁)
2 simp1 1130 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝐾 ∈ HL)
3 eqid 2819 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
4 llncmp.n . . . . . . 7 𝑁 = (LLines‘𝐾)
53, 4llnbase 36637 . . . . . 6 (𝑋𝑁𝑋 ∈ (Base‘𝐾))
653ad2ant2 1128 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋 ∈ (Base‘𝐾))
7 eqid 2819 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8 eqid 2819 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
93, 7, 8, 4islln4 36635 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
102, 6, 9syl2anc 586 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
111, 10mpbid 234 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋)
12 simpr3 1190 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 𝑌)
13 hlpos 36494 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Poset)
14133ad2ant1 1127 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝐾 ∈ Poset)
1514adantr 483 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝐾 ∈ Poset)
166adantr 483 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17 simpl3 1187 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑌𝑁)
183, 4llnbase 36637 . . . . . . . 8 (𝑌𝑁𝑌 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑌 ∈ (Base‘𝐾))
20 simpr1 1188 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
213, 8atbase 36417 . . . . . . . 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 1189 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
24 simpl1 1185 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝐾 ∈ HL)
25 llncmp.l . . . . . . . . . . 11 = (le‘𝐾)
263, 25, 7cvrle 36406 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑝( ⋖ ‘𝐾)𝑋) → 𝑝 𝑋)
2724, 22, 16, 23, 26syl31anc 1367 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 𝑋)
283, 25postr 17555 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
2915, 22, 16, 19, 28syl13anc 1366 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
3027, 12, 29mp2and 697 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 𝑌)
3125, 7, 8, 4atcvrlln2 36647 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑌𝑁) ∧ 𝑝 𝑌) → 𝑝( ⋖ ‘𝐾)𝑌)
3224, 20, 17, 30, 31syl31anc 1367 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑌)
333, 25, 7cvrcmp 36411 . . . . . . 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 1382 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
3512, 34mpbid 234 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 = 𝑌)
36353exp2 1348 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝( ⋖ ‘𝐾)𝑋 → (𝑋 𝑌𝑋 = 𝑌))))
3736rexlimdv 3281 . . 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 17553 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 𝑋)
4014, 6, 39syl2anc 586 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋 𝑋)
41 breq2 5061 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
4240, 41syl5ibcom 247 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 = 𝑌𝑋 𝑌))
4338, 42impbid 214 1 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wrex 3137   class class class wbr 5057  cfv 6348  Basecbs 16475  lecple 16564  Posetcpo 17542  ccvr 36390  Atomscatm 36391  HLchlt 36478  LLinesclln 36619
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 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-llines 36626
This theorem is referenced by:  llnnlt  36651  2llnmat  36652  llnmlplnN  36667  dalem16  36807  dalem60  36860  llnexchb2  36997
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