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Theorem cvrexch 29609
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 22949 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 29608 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) C Y  ->  X C ( X  .\/  Y ) ) )
6 simp1 955 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
7 hlop 29552 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
873ad2ant1 976 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
9 simp3 957 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
10 eqid 2283 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
111, 10opoccl 29384 . . . . . 6  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
128, 9, 11syl2anc 642 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
13 simp2 956 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
141, 10opoccl 29384 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
158, 13, 14syl2anc 642 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
161, 2, 3, 4cvrexchlem 29608 . . . . 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 1182 . . . 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 29551 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
191, 2, 3, 10oldmj1 29411 . . . . . . 7  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) ) )
2018, 19syl3an1 1215 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) ) )
21 hllat 29553 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
22213ad2ant1 976 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
231, 3latmcom 14181 . . . . . . 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 1182 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
)  =  ( ( ( oc `  K
) `  Y )  ./\  ( ( oc `  K ) `  X
) ) )
2520, 24eqtrd 2315 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 Y )  ./\  ( ( oc `  K ) `  X
) ) )
2625breq1d 4033 . . . 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 29407 . . . . . . 7  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) ) )
2818, 27syl3an1 1215 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) ) )
291, 2latjcom 14165 . . . . . . 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 1182 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X )  .\/  (
( oc `  K
) `  Y )
)  =  ( ( ( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  X
) ) )
3128, 30eqtrd 2315 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 Y )  .\/  ( ( oc `  K ) `  X
) ) )
3231breq2d 4035 . . . 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 259 . . 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 14156 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
3521, 34syl3an1 1215 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
361, 10, 4cvrcon3b 29467 . . . 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 1182 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( X  .\/  Y )  <-> 
( ( oc `  K ) `  ( X  .\/  Y ) ) C ( ( oc
`  K ) `  X ) ) )
381, 3latmcl 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
3921, 38syl3an1 1215 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
401, 10, 4cvrcon3b 29467 . . . 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 1182 . . 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 259 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( X  .\/  Y )  ->  ( X  ./\  Y ) C Y ) )
435, 42impbid 183 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 176    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   OLcol 29364    <o ccvr 29452   HLchlt 29540
This theorem is referenced by:  cvrat3  29631  2lplnmN  29748  2llnmj  29749  2llnm2N  29757  2lplnm2N  29810  2lplnmj  29811  lhpmcvr  30212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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