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Theorem cvrexch 32904
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 25724 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 32903 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) C Y  ->  X C ( X  .\/  Y ) ) )
6 simp1 988 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
7 hlop 32847 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
873ad2ant1 1009 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
9 simp3 990 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
10 eqid 2438 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
111, 10opoccl 32679 . . . . . 6  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
128, 9, 11syl2anc 661 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
13 simp2 989 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
141, 10opoccl 32679 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
158, 13, 14syl2anc 661 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
161, 2, 3, 4cvrexchlem 32903 . . . . 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 1218 . . . 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 32846 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
191, 2, 3, 10oldmj1 32706 . . . . . . 7  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) ) )
2018, 19syl3an1 1251 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) ) )
21 hllat 32848 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
22213ad2ant1 1009 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
231, 3latmcom 15237 . . . . . . 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 1218 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
)  =  ( ( ( oc `  K
) `  Y )  ./\  ( ( oc `  K ) `  X
) ) )
2520, 24eqtrd 2470 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 Y )  ./\  ( ( oc `  K ) `  X
) ) )
2625breq1d 4297 . . . 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 32702 . . . . . . 7  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) ) )
2818, 27syl3an1 1251 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) ) )
291, 2latjcom 15221 . . . . . . 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 1218 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X )  .\/  (
( oc `  K
) `  Y )
)  =  ( ( ( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  X
) ) )
3128, 30eqtrd 2470 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 Y )  .\/  ( ( oc `  K ) `  X
) ) )
3231breq2d 4299 . . . 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 268 . . 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 15213 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
3521, 34syl3an1 1251 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
361, 10, 4cvrcon3b 32762 . . . 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 1218 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( X  .\/  Y )  <-> 
( ( oc `  K ) `  ( X  .\/  Y ) ) C ( ( oc
`  K ) `  X ) ) )
381, 3latmcl 15214 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
3921, 38syl3an1 1251 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
401, 10, 4cvrcon3b 32762 . . . 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 1218 . . 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 268 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( X  .\/  Y )  ->  ( X  ./\  Y ) C Y ) )
435, 42impbid 191 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) C Y  <->  X C
( X  .\/  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   occoc 14238   joincjn 15106   meetcmee 15107   Latclat 15207   OPcops 32657   OLcol 32659    <o ccvr 32747   HLchlt 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836
This theorem is referenced by:  cvrat3  32926  2lplnmN  33043  2llnmj  33044  2llnm2N  33052  2lplnm2N  33105  2lplnmj  33106  lhpmcvr  33507
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