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Theorem lvolcmp 29806
Description: If two lattice planes are comparable, they are equal. (Contributed by NM, 12-Jul-2012.)
Hypotheses
Ref Expression
lvolcmp.l  |-  .<_  =  ( le `  K )
lvolcmp.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvolcmp  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .<_  Y  <->  X  =  Y ) )

Proof of Theorem lvolcmp
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simp2 956 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  X  e.  V )
2 simp1 955 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  K  e.  HL )
3 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
4 lvolcmp.v . . . . . . 7  |-  V  =  ( LVols `  K )
53, 4lvolbase 29767 . . . . . 6  |-  ( X  e.  V  ->  X  e.  ( Base `  K
) )
653ad2ant2 977 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  X  e.  ( Base `  K ) )
7 eqid 2283 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
8 eqid 2283 . . . . . 6  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
93, 7, 8, 4islvol4 29763 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K ) )  -> 
( X  e.  V  <->  E. z  e.  ( LPlanes `  K ) z ( 
<o  `  K ) X ) )
102, 6, 9syl2anc 642 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  e.  V  <->  E. z  e.  ( LPlanes `  K ) z ( 
<o  `  K ) X ) )
111, 10mpbid 201 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  E. z  e.  (
LPlanes `  K ) z (  <o  `  K ) X )
12 simpr3 963 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  ->  X  .<_  Y )
13 hlpos 29555 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Poset )
14133ad2ant1 976 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  K  e.  Poset )
1514adantr 451 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  ->  K  e.  Poset )
166adantr 451 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  ->  X  e.  ( Base `  K ) )
17 simpl3 960 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  ->  Y  e.  V )
183, 4lvolbase 29767 . . . . . . . 8  |-  ( Y  e.  V  ->  Y  e.  ( Base `  K
) )
1917, 18syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  ->  Y  e.  ( Base `  K ) )
20 simpr1 961 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  -> 
z  e.  ( LPlanes `  K ) )
213, 8lplnbase 29723 . . . . . . . 8  |-  ( z  e.  ( LPlanes `  K
)  ->  z  e.  ( Base `  K )
)
2220, 21syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  -> 
z  e.  ( Base `  K ) )
23 simpr2 962 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  -> 
z (  <o  `  K
) X )
24 simpl1 958 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  ->  K  e.  HL )
25 lvolcmp.l . . . . . . . . . . 11  |-  .<_  =  ( le `  K )
263, 25, 7cvrle 29468 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  z  e.  ( Base `  K )  /\  X  e.  ( Base `  K
) )  /\  z
(  <o  `  K ) X )  ->  z  .<_  X )
2724, 22, 16, 23, 26syl31anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  -> 
z  .<_  X )
283, 25postr 14087 . . . . . . . . . 10  |-  ( ( K  e.  Poset  /\  (
z  e.  ( Base `  K )  /\  X  e.  ( Base `  K
)  /\  Y  e.  ( Base `  K )
) )  ->  (
( z  .<_  X  /\  X  .<_  Y )  -> 
z  .<_  Y ) )
2915, 22, 16, 19, 28syl13anc 1184 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  -> 
( ( z  .<_  X  /\  X  .<_  Y )  ->  z  .<_  Y ) )
3027, 12, 29mp2and 660 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  -> 
z  .<_  Y )
3125, 7, 8, 4lplncvrlvol2 29804 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  z  e.  ( LPlanes `  K )  /\  Y  e.  V )  /\  z  .<_  Y )  ->  z
(  <o  `  K ) Y )
3224, 20, 17, 30, 31syl31anc 1185 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  -> 
z (  <o  `  K
) Y )
333, 25, 7cvrcmp 29473 . . . . . . 7  |-  ( ( K  e.  Poset  /\  ( X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
)  /\  z  e.  ( Base `  K )
)  /\  ( z
(  <o  `  K ) X  /\  z (  <o  `  K ) Y ) )  ->  ( X  .<_  Y  <->  X  =  Y
) )
3415, 16, 19, 22, 23, 32, 33syl132anc 1200 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  -> 
( X  .<_  Y  <->  X  =  Y ) )
3512, 34mpbid 201 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  /\  ( z  e.  (
LPlanes `  K )  /\  z (  <o  `  K
) X  /\  X  .<_  Y ) )  ->  X  =  Y )
36353exp2 1169 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  ( z  e.  (
LPlanes `  K )  -> 
( z (  <o  `  K ) X  -> 
( X  .<_  Y  ->  X  =  Y )
) ) )
3736rexlimdv 2666 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  ( E. z  e.  ( LPlanes `  K )
z (  <o  `  K
) X  ->  ( X  .<_  Y  ->  X  =  Y ) ) )
3811, 37mpd 14 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .<_  Y  ->  X  =  Y )
)
393, 25posref 14085 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  ( Base `  K
) )  ->  X  .<_  X )
4014, 6, 39syl2anc 642 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  X  .<_  X )
41 breq2 4027 . . 3  |-  ( X  =  Y  ->  ( X  .<_  X  <->  X  .<_  Y ) )
4240, 41syl5ibcom 211 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  =  Y  ->  X  .<_  Y ) )
4338, 42impbid 183 1  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .<_  Y  <->  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074    <o ccvr 29452   HLchlt 29540   LPlanesclpl 29681   LVolsclvol 29682
This theorem is referenced by:  lvolnltN  29807  2lplnja  29808
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-3or 935  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  df-llines 29687  df-lplanes 29688  df-lvols 29689
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