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Theorem cvrcmp 35042
Description: If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
Hypotheses
Ref Expression
cvrcmp.b 𝐵 = (Base‘𝐾)
cvrcmp.l = (le‘𝐾)
cvrcmp.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrcmp ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))

Proof of Theorem cvrcmp
StepHypRef Expression
1 simpl1 1204 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝐾 ∈ Poset)
2 simpl23 1301 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐵)
3 simpl21 1297 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋𝐵)
4 simpl3l 1263 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑋)
5 cvrcmp.b . . . . . 6 𝐵 = (Base‘𝐾)
6 cvrcmp.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
75, 6cvrne 35040 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍𝑋)
81, 2, 3, 4, 7syl31anc 1466 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝑋)
9 cvrcmp.l . . . . . . . 8 = (le‘𝐾)
105, 9, 6cvrle 35037 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑍𝐵𝑋𝐵) ∧ 𝑍𝐶𝑋) → 𝑍 𝑋)
111, 2, 3, 4, 10syl31anc 1466 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍 𝑋)
12 simpr 479 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 𝑌)
13 simpl22 1299 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑌𝐵)
14 simpl3r 1265 . . . . . . 7 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑍𝐶𝑌)
155, 9, 6cvrnbtwn4 35038 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑍𝐵𝑌𝐵𝑋𝐵) ∧ 𝑍𝐶𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
161, 2, 13, 3, 14, 15syl131anc 1476 . . . . . 6 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → ((𝑍 𝑋𝑋 𝑌) ↔ (𝑍 = 𝑋𝑋 = 𝑌)))
1711, 12, 16mpbi2and 994 . . . . 5 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍 = 𝑋𝑋 = 𝑌))
18 neor 3011 . . . . 5 ((𝑍 = 𝑋𝑋 = 𝑌) ↔ (𝑍𝑋𝑋 = 𝑌))
1917, 18sylib 208 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → (𝑍𝑋𝑋 = 𝑌))
208, 19mpd 15 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) ∧ 𝑋 𝑌) → 𝑋 = 𝑌)
2120ex 449 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
22 simp1 1128 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝐾 ∈ Poset)
23 simp21 1225 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋𝐵)
245, 9posref 17123 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
2522, 23, 24syl2anc 696 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → 𝑋 𝑋)
26 breq2 4796 . . 3 (𝑋 = 𝑌 → (𝑋 𝑋𝑋 𝑌))
2725, 26syl5ibcom 235 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 = 𝑌𝑋 𝑌))
2821, 27impbid 202 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920   class class class wbr 4792  cfv 6037  Basecbs 16030  lecple 16121  Posetcpo 17112  ccvr 35021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-iota 6000  df-fun 6039  df-fv 6045  df-preset 17100  df-poset 17118  df-plt 17130  df-covers 35025
This theorem is referenced by:  cvrcmp2  35043  atcmp  35070  llncmp  35280  lplncmp  35320  lvolcmp  35375  lhp2lt  35759
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