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Theorem cvrexch 35541
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 27486 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 35540 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y ) C Y  ->  X C ( X  .\/  Y ) ) )
6 simp1 994 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
7 hlop 35484 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
873ad2ant1 1015 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
9 simp3 996 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
10 eqid 2454 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
111, 10opoccl 35316 . . . . . 6  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
128, 9, 11syl2anc 659 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  Y
)  e.  B )
13 simp2 995 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
141, 10opoccl 35316 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
158, 13, 14syl2anc 659 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
161, 2, 3, 4cvrexchlem 35540 . . . . 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 1226 . . . 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 35483 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
191, 2, 3, 10oldmj1 35343 . . . . . . 7  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) ) )
2018, 19syl3an1 1259 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 X )  ./\  ( ( oc `  K ) `  Y
) ) )
21 hllat 35485 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
22213ad2ant1 1015 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
231, 3latmcom 15904 . . . . . . 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 1226 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X )  ./\  (
( oc `  K
) `  Y )
)  =  ( ( ( oc `  K
) `  Y )  ./\  ( ( oc `  K ) `  X
) ) )
2520, 24eqtrd 2495 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  .\/  Y ) )  =  ( ( ( oc `  K ) `
 Y )  ./\  ( ( oc `  K ) `  X
) ) )
2625breq1d 4449 . . . 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 35339 . . . . . . 7  |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) ) )
2818, 27syl3an1 1259 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 X )  .\/  ( ( oc `  K ) `  Y
) ) )
291, 2latjcom 15888 . . . . . . 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 1226 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X )  .\/  (
( oc `  K
) `  Y )
)  =  ( ( ( oc `  K
) `  Y )  .\/  ( ( oc `  K ) `  X
) ) )
3128, 30eqtrd 2495 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  ( X  ./\  Y ) )  =  ( ( ( oc `  K ) `
 Y )  .\/  ( ( oc `  K ) `  X
) ) )
3231breq2d 4451 . . . 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 15880 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
3521, 34syl3an1 1259 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
361, 10, 4cvrcon3b 35399 . . . 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 1226 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C ( X  .\/  Y )  <-> 
( ( oc `  K ) `  ( X  .\/  Y ) ) C ( ( oc
`  K ) `  X ) ) )
381, 3latmcl 15881 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
3921, 38syl3an1 1259 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
401, 10, 4cvrcon3b 35399 . . . 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 1226 . . 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 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   occoc 14792   joincjn 15772   meetcmee 15773   Latclat 15874   OPcops 35294   OLcol 35296    <o ccvr 35384   HLchlt 35472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473
This theorem is referenced by:  cvrat3  35563  2lplnmN  35680  2llnmj  35681  2llnm2N  35689  2lplnm2N  35742  2lplnmj  35743  lhpmcvr  36144
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