Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrcmp2 Structured version   Visualization version   Unicode version
Theorem cvrcmp2 34571
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 = ( <o ` K )
Assertion
Ref Expression
cvrcmp2 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( X .<_ Y <-> X = Y ) )
Proof of Theorem cvrcmp2
StepHypRef Expression
1 opposet 34468 . . . 4 |- ( K e. OP -> K e. Poset )
213ad2ant1 1082 . . 3 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> K e. Poset )
3 simp1 1061 . . . 4 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> K e. OP )
4 simp22 1095 . . . 4 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> Y e. B )
5 cvrcmp.b . . . . 5 |- B = ( Base ` K )
6 eqid 2622 . . . . 5 |- ( oc ` K ) = ( oc ` K )
75, 6opoccl 34481 . . . 4 |- ( ( K e. OP /\ Y e. B ) -> ( ( oc ` K ) ` Y ) e. B )
83, 4, 7syl2anc 693 . . 3 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( ( oc ` K ) ` Y ) e. B )
9 simp21 1094 . . . 4 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> X e. B )
105, 6opoccl 34481 . . . 4 |- ( ( K e. OP /\ X e. B ) -> ( ( oc ` K ) ` X ) e. B )
113, 9, 10syl2anc 693 . . 3 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( ( oc ` K ) ` X ) e. B )
12 simp23 1096 . . . 4 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> Z e. B )
135, 6opoccl 34481 . . . 4 |- ( ( K e. OP /\ Z e. B ) -> ( ( oc ` K ) ` Z ) e. B )
143, 12, 13syl2anc 693 . . 3 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( ( oc ` K ) ` Z ) e. B )
15 cvrcmp.c . . . . . . . 8 |- C = ( <o ` K )
165, 6, 15cvrcon3b 34564 . . . . . . 7 |- ( ( K e. OP /\ X e. B /\ Z e. B ) -> ( X C Z <-> ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` X ) ) )
17163adant3r2 1275 . . . . . 6 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Z <-> ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` X ) ) )
185, 6, 15cvrcon3b 34564 . . . . . . 7 |- ( ( K e. OP /\ Y e. B /\ Z e. B ) -> ( Y C Z <-> ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` Y ) ) )
19183adant3r1 1274 . . . . . 6 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y C Z <-> ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` Y ) ) )
2017, 19anbi12d 747 . . . . 5 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X C Z /\ Y C Z ) <-> ( ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` X ) /\ ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` Y ) ) ) )
2120biimp3a 1432 . . . 4 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` X ) /\ ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` Y ) ) )
2221ancomd 467 . . 3 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` Y ) /\ ( ( oc ` K ) ` Z ) C ( ( oc ` K ) ` X ) ) )
23 cvrcmp.l . . . 4 |- .<_ = ( le ` K )
245, 23, 15cvrcmp 34570 . . 3 |- ( ( K e. Poset /\ ( ( ( oc ` K ) ` Y ) e. B /\ ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Z ) e. 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 1339 . 2 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( ( ( oc ` K ) ` Y ) .<_ ( ( oc ` K ) ` X ) <-> ( ( oc ` K ) ` Y ) = ( ( oc ` K ) ` X ) ) )
265, 23, 6oplecon3b 34487 . . 3 |- ( ( K e. OP /\ X e. B /\ Y e. B ) -> ( X .<_ Y <-> ( ( oc ` K ) ` Y ) .<_ ( ( oc ` K ) ` X ) ) )
273, 9, 4, 26syl3anc 1326 . 2 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( X .<_ Y <-> ( ( oc ` K ) ` Y ) .<_ ( ( oc ` K ) ` X ) ) )
285, 6opcon3b 34483 . . 3 |- ( ( K e. OP /\ X e. B /\ Y e. B ) -> ( X = Y <-> ( ( oc ` K ) ` Y ) = ( ( oc ` K ) ` X ) ) )
293, 9, 4, 28syl3anc 1326 . 2 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( X = Y <-> ( ( oc ` K ) ` Y ) = ( ( oc ` K ) ` X ) ) )
3025, 27, 293bitr4d 300 1 |- ( ( K e. OP /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Z /\ Y C Z ) ) -> ( X .<_ Y <-> X = Y ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   occoc 15949   Posetcpo 16940   OPcops 34459   <o ccvr 34549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-preset 16928  df-poset 16946  df-plt 16958  df-oposet 34463  df-covers 34553
This theorem is referenced by:  llncvrlpln  34844  lplncvrlvol  34902
  Copyright terms: Public domain W3C validator