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Theorem cvrcmp2 38693
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 38590 . . . 4 (𝐾 ∈ OP → 𝐾 ∈ Poset)
213ad2ant1 1131 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Poset)
3 simp1 1134 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OP)
4 simp22 1205 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 cvrcmp.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2727 . . . . 5 (oc‘𝐾) = (oc‘𝐾)
75, 6opoccl 38603 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
83, 4, 7syl2anc 583 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
9 simp21 1204 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
105, 6opoccl 38603 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
113, 9, 10syl2anc 583 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
12 simp23 1206 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
135, 6opoccl 38603 . . . 4 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
143, 12, 13syl2anc 583 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
15 cvrcmp.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
165, 6, 15cvrcon3b 38686 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
17163adant3r2 1181 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
185, 6, 15cvrcon3b 38686 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
19183adant3r1 1180 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2017, 19anbi12d 630 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑍𝑌𝐶𝑍) ↔ (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌))))
2120biimp3a 1466 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2221ancomd 461 . . 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 38692 . . 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 1381 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋) ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
265, 23, 6oplecon3b 38609 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
273, 9, 4, 26syl3anc 1369 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
285, 6opcon3b 38605 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
293, 9, 4, 28syl3anc 1369 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
3025, 27, 293bitr4d 311 1 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099   class class class wbr 5142  cfv 6542  Basecbs 17171  lecple 17231  occoc 17232  Posetcpo 18290  OPcops 38581  ccvr 38671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-proset 18278  df-poset 18296  df-plt 18313  df-oposet 38585  df-covers 38675
This theorem is referenced by:  llncvrlpln  38968  lplncvrlvol  39026
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