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Theorem cvrexch 29535
Description: A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 23721 analog.) (Contributed by NM, 18-Nov-2011.)
Hypotheses
Ref Expression
cvrexch.b  |-  B  =  ( Base `  K
)
cvrexch.j  |-  .\/  =  ( join `  K )
cvrexch.m  |-  ./\  =  ( meet `  K )
cvrexch.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrexch  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) C Y  <->  X C
( X  .\/  Y
) ) )

Proof of Theorem cvrexch
StepHypRef Expression
1 cvrexch.b . . 3  |-  B  =  ( Base `  K
)
2 cvrexch.j . . 3  |-  .\/  =  ( join `  K )
3 cvrexch.m . . 3  |-  ./\  =  ( meet `  K )
4 cvrexch.c . . 3  |-  C  =  (  <o  `  K )
51, 2, 3, 4cvrexchlem 29534 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) C Y  ->  X C ( X  .\/  Y ) ) )
6 simp1 957 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
7 hlop 29478 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
873ad2ant1 978 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
9 simp3 959 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
10 eqid 2388 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
111, 10opoccl 29310 . . . . . 6  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
128, 9, 11syl2anc 643 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
13 simp2 958 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
141, 10opoccl 29310 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
158, 13, 14syl2anc 643 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
161, 2, 3, 4cvrexchlem 29534 . . . . 5  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  Y
)  e.  B  /\  ( ( oc `  K ) `  X
)  e.  B )  ->  ( ( ( ( oc `  K
) `  Y )  ./\  ( ( oc `  K ) `  X
) ) C ( ( oc `  K
) `  X )  ->  ( ( oc `  K ) `  Y
) C ( ( ( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  X
) ) ) )
176, 12, 15, 16syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( ( oc `  K ) `
 Y )  ./\  ( ( oc `  K ) `  X
) ) C ( ( oc `  K
) `  X )  ->  ( ( oc `  K ) `  Y
) C ( ( ( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  X
) ) ) )
18 hlol 29477 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
191, 2, 3, 10oldmj1 29337 . . . . . . 7  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) ) )
2018, 19syl3an1 1217 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) ) )
21 hllat 29479 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
22213ad2ant1 978 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
231, 3latmcom 14432 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B )  ->  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) )  =  ( ( ( oc `  K ) `  Y
)  ./\  ( ( oc `  K ) `  X ) ) )
2422, 15, 12, 23syl3anc 1184 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
)  =  ( ( ( oc `  K
) `  Y )  ./\  ( ( oc `  K ) `  X
) ) )
2520, 24eqtrd 2420 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 Y )  ./\  ( ( oc `  K ) `  X
) ) )
2625breq1d 4164 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  ( X  .\/  Y ) ) C ( ( oc `  K ) `
 X )  <->  ( (
( oc `  K
) `  Y )  ./\  ( ( oc `  K ) `  X
) ) C ( ( oc `  K
) `  X )
) )
271, 2, 3, 10oldmm1 29333 . . . . . . 7  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) ) )
2818, 27syl3an1 1217 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) ) )
291, 2latjcom 14416 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  ( ( oc `  K ) `  Y
)  e.  B )  ->  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) )  =  ( ( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  X ) ) )
3022, 15, 12, 29syl3anc 1184 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X )  .\/  (
( oc `  K
) `  Y )
)  =  ( ( ( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  X
) ) )
3128, 30eqtrd 2420 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 Y )  .\/  ( ( oc `  K ) `  X
) ) )
3231breq2d 4166 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  Y ) C ( ( oc `  K
) `  ( X  ./\ 
Y ) )  <->  ( ( oc `  K ) `  Y ) C ( ( ( oc `  K ) `  Y
)  .\/  ( ( oc `  K ) `  X ) ) ) )
3317, 26, 323imtr4d 260 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  ( X  .\/  Y ) ) C ( ( oc `  K ) `
 X )  -> 
( ( oc `  K ) `  Y
) C ( ( oc `  K ) `
 ( X  ./\  Y ) ) ) )
341, 2latjcl 14407 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
3521, 34syl3an1 1217 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
361, 10, 4cvrcon3b 29393 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X C ( X  .\/  Y )  <-> 
( ( oc `  K ) `  ( X  .\/  Y ) ) C ( ( oc
`  K ) `  X ) ) )
378, 13, 35, 36syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( X  .\/  Y )  <-> 
( ( oc `  K ) `  ( X  .\/  Y ) ) C ( ( oc
`  K ) `  X ) ) )
381, 3latmcl 14408 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
3921, 38syl3an1 1217 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
401, 10, 4cvrcon3b 29393 . . . 4  |-  ( ( K  e.  OP  /\  ( X  ./\  Y )  e.  B  /\  Y  e.  B )  ->  (
( X  ./\  Y
) C Y  <->  ( ( oc `  K ) `  Y ) C ( ( oc `  K
) `  ( X  ./\ 
Y ) ) ) )
418, 39, 9, 40syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) C Y  <->  ( ( oc `  K ) `  Y ) C ( ( oc `  K
) `  ( X  ./\ 
Y ) ) ) )
4233, 37, 413imtr4d 260 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( X  .\/  Y )  ->  ( X  ./\  Y ) C Y ) )
435, 42impbid 184 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) C Y  <->  X C
( X  .\/  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   occoc 13465   joincjn 14329   meetcmee 14330   Latclat 14402   OPcops 29288   OLcol 29290    <o ccvr 29378   HLchlt 29466
This theorem is referenced by:  cvrat3  29557  2lplnmN  29674  2llnmj  29675  2llnm2N  29683  2lplnm2N  29736  2lplnmj  29737  lhpmcvr  30138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467
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