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Theorem cvrcmp2 34086
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 33983 . . . 4 (𝐾 ∈ OP → 𝐾 ∈ Poset)
213ad2ant1 1080 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ Poset)
3 simp1 1059 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝐾 ∈ OP)
4 simp22 1093 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑌𝐵)
5 cvrcmp.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2621 . . . . 5 (oc‘𝐾) = (oc‘𝐾)
75, 6opoccl 33996 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
83, 4, 7syl2anc 692 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
9 simp21 1092 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑋𝐵)
105, 6opoccl 33996 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
113, 9, 10syl2anc 692 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
12 simp23 1094 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → 𝑍𝐵)
135, 6opoccl 33996 . . . 4 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
143, 12, 13syl2anc 692 . . 3 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
15 cvrcmp.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
165, 6, 15cvrcon3b 34079 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
17163adant3r2 1272 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋)))
185, 6, 15cvrcon3b 34079 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
19183adant3r1 1271 . . . . . 6 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌𝐶𝑍 ↔ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2017, 19anbi12d 746 . . . . 5 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑍𝑌𝐶𝑍) ↔ (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌))))
2120biimp3a 1429 . . . 4 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑋) ∧ ((oc‘𝐾)‘𝑍)𝐶((oc‘𝐾)‘𝑌)))
2221ancomd 467 . . 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 34085 . . 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 1336 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋) ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
265, 23, 6oplecon3b 34002 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
273, 9, 4, 26syl3anc 1323 . 2 ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌 ↔ ((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑋)))
285, 6opcon3b 33998 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ((oc‘𝐾)‘𝑌) = ((oc‘𝐾)‘𝑋)))
293, 9, 4, 28syl3anc 1323 . 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 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4618  cfv 5852  Basecbs 15792  lecple 15880  occoc 15881  Posetcpo 16872  OPcops 33974  ccvr 34064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-preset 16860  df-poset 16878  df-plt 16890  df-oposet 33978  df-covers 34068
This theorem is referenced by:  llncvrlpln  34359  lplncvrlvol  34417
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