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Theorem llncmp 35329
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 1132 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋𝑁)
2 simp1 1131 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝐾 ∈ HL)
3 eqid 2760 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
4 llncmp.n . . . . . . 7 𝑁 = (LLines‘𝐾)
53, 4llnbase 35316 . . . . . 6 (𝑋𝑁𝑋 ∈ (Base‘𝐾))
653ad2ant2 1129 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋 ∈ (Base‘𝐾))
7 eqid 2760 . . . . . 6 ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8 eqid 2760 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
93, 7, 8, 4islln4 35314 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
102, 6, 9syl2anc 696 . . . 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 1238 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑋 𝑌)
13 hlpos 35173 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Poset)
14133ad2ant1 1128 . . . . . . . 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 1232 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑌𝑁)
183, 4llnbase 35316 . . . . . . . 8 (𝑌𝑁𝑌 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑌 ∈ (Base‘𝐾))
20 simpr1 1234 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
213, 8atbase 35097 . . . . . . . 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 1236 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
24 simpl1 1228 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝐾 ∈ HL)
25 llncmp.l . . . . . . . . . . 11 = (le‘𝐾)
263, 25, 7cvrle 35086 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑝( ⋖ ‘𝐾)𝑋) → 𝑝 𝑋)
2724, 22, 16, 23, 26syl31anc 1480 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 𝑋)
283, 25postr 17174 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
2915, 22, 16, 19, 28syl13anc 1479 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → ((𝑝 𝑋𝑋 𝑌) → 𝑝 𝑌))
3027, 12, 29mp2and 717 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝 𝑌)
3125, 7, 8, 4atcvrlln2 35326 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑌𝑁) ∧ 𝑝 𝑌) → 𝑝( ⋖ ‘𝐾)𝑌)
3224, 20, 17, 30, 31syl31anc 1480 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋𝑋 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑌)
333, 25, 7cvrcmp 35091 . . . . . . 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 1495 . . . . . 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 1448 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝( ⋖ ‘𝐾)𝑋 → (𝑋 𝑌𝑋 = 𝑌))))
3736rexlimdv 3168 . . 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 17172 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 𝑋)
4014, 6, 39syl2anc 696 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) → 𝑋 𝑋)
41 breq2 4808 . . 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 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051   class class class wbr 4804  cfv 6049  Basecbs 16079  lecple 16170  Posetcpo 17161  ccvr 35070  Atomscatm 35071  HLchlt 35158  LLinesclln 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-llines 35305
This theorem is referenced by:  llnnlt  35330  2llnmat  35331  llnmlplnN  35346  dalem16  35486  dalem60  35539  llnexchb2  35676
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