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

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
 Copyright terms: Public domain W3C validator