Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrcmp2 Structured version   Visualization version   GIF version

Theorem cvrcmp2 35092
Description: If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
Hypotheses
Ref Expression
cvrcmp.b 𝐵 = (Base‘𝐾)
cvrcmp.l = (le‘𝐾)
cvrcmp.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrcmp2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌𝑋 = 𝑌))

Proof of Theorem cvrcmp2
StepHypRef Expression
1 opposet 34989 . . . 4 (𝐾 ∈ OP → 𝐾 ∈ Poset)
213ad2ant1 1128 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Poset)
3 simp1 1131 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OP)
4 simp22 1250 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 cvrcmp.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2760 . . . . 5 (oc‘𝐾) = (oc‘𝐾)
75, 6opoccl 35002 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
83, 4, 7syl2anc 696 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
9 simp21 1249 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
105, 6opoccl 35002 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
113, 9, 10syl2anc 696 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
12 simp23 1251 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
135, 6opoccl 35002 . . . 4 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
143, 12, 13syl2anc 696 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
15 cvrcmp.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
165, 6, 15cvrcon3b 35085 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
17163adant3r2 1199 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
185, 6, 15cvrcon3b 35085 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
19183adant3r1 1198 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2017, 19anbi12d 749 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑍𝑌𝐶𝑍) ↔ (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌))))
2120biimp3a 1581 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2221ancomd 466 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
23 cvrcmp.l . . . 4 = (le‘𝐾)
245, 23, 15cvrcmp 35091 . . 3 ((𝐾 ∈ Poset ∧ (((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) ∧ (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋))) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋) ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
252, 8, 11, 14, 22, 24syl131anc 1490 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋) ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
265, 23, 6oplecon3b 35008 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
273, 9, 4, 26syl3anc 1477 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
285, 6opcon3b 35004 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
293, 9, 4, 28syl3anc 1477 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
3025, 27, 293bitr4d 300 1 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  Basecbs 16079  lecple 16170  occoc 16171  Posetcpo 17161  OPcops 34980  ccvr 35070
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-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-rab 3059  df-v 3342  df-sbc 3577  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-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-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-preset 17149  df-poset 17167  df-plt 17179  df-oposet 34984  df-covers 35074
This theorem is referenced by:  llncvrlpln  35365  lplncvrlvol  35423
  Copyright terms: Public domain W3C validator