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Theorem cvrcmp2 35261
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 35158 . . . 4 (𝐾 ∈ OP → 𝐾 ∈ Poset)
213ad2ant1 1163 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Poset)
3 simp1 1166 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OP)
4 simp22 1264 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 cvrcmp.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2765 . . . . 5 (oc‘𝐾) = (oc‘𝐾)
75, 6opoccl 35171 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
83, 4, 7syl2anc 579 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
9 simp21 1263 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
105, 6opoccl 35171 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
113, 9, 10syl2anc 579 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
12 simp23 1265 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
135, 6opoccl 35171 . . . 4 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
143, 12, 13syl2anc 579 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
15 cvrcmp.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
165, 6, 15cvrcon3b 35254 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
17163adant3r2 1234 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
185, 6, 15cvrcon3b 35254 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
19183adant3r1 1233 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2017, 19anbi12d 624 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑍𝑌𝐶𝑍) ↔ (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌))))
2120biimp3a 1593 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2221ancomd 453 . . 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 35260 . . 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 1502 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋) ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
265, 23, 6oplecon3b 35177 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
273, 9, 4, 26syl3anc 1490 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
285, 6opcon3b 35173 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
293, 9, 4, 28syl3anc 1490 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
3025, 27, 293bitr4d 302 1 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155   class class class wbr 4811  cfv 6070  Basecbs 16144  lecple 16235  occoc 16236  Posetcpo 17220  OPcops 35149  ccvr 35239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078  df-ov 6849  df-proset 17208  df-poset 17226  df-plt 17238  df-oposet 35153  df-covers 35243
This theorem is referenced by:  llncvrlpln  35535  lplncvrlvol  35593
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