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Theorem cvrcmp2 33589
 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 33486 . . . 4 (𝐾 ∈ OP → 𝐾 ∈ Poset)
213ad2ant1 1075 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Poset)
3 simp1 1054 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OP)
4 simp22 1088 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 cvrcmp.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2610 . . . . 5 (oc‘𝐾) = (oc‘𝐾)
75, 6opoccl 33499 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
83, 4, 7syl2anc 691 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
9 simp21 1087 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
105, 6opoccl 33499 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
113, 9, 10syl2anc 691 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
12 simp23 1089 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
135, 6opoccl 33499 . . . 4 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
143, 12, 13syl2anc 691 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
15 cvrcmp.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
165, 6, 15cvrcon3b 33582 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
17163adant3r2 1267 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
185, 6, 15cvrcon3b 33582 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
19183adant3r1 1266 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2017, 19anbi12d 743 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑍𝑌𝐶𝑍) ↔ (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌))))
2120biimp3a 1424 . . . 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 33588 . . 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 1331 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋) ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
265, 23, 6oplecon3b 33505 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
273, 9, 4, 26syl3anc 1318 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
285, 6opcon3b 33501 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
293, 9, 4, 28syl3anc 1318 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
3025, 27, 293bitr4d 299 1 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  occoc 15776  Posetcpo 16763  OPcops 33477   ⋖ ccvr 33567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-preset 16751  df-poset 16769  df-plt 16781  df-oposet 33481  df-covers 33571 This theorem is referenced by:  llncvrlpln  33862  lplncvrlvol  33920
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