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Theorem pmap11 33429
Description: The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
pmap11.b  |-  B  =  ( Base `  K
)
pmap11.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmap11  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( M `  X )  =  ( M `  Y )  <-> 
X  =  Y ) )

Proof of Theorem pmap11
StepHypRef Expression
1 hllat 33031 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
2 pmap11.b . . . . 5  |-  B  =  ( Base `  K
)
3 eqid 2443 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
42, 3latasymb 15243 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X ( le `  K ) Y  /\  Y ( le `  K ) X )  <->  X  =  Y ) )
51, 4syl3an1 1251 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X ( le `  K ) Y  /\  Y ( le `  K ) X )  <->  X  =  Y ) )
6 pmap11.m . . . . 5  |-  M  =  ( pmap `  K
)
72, 3, 6pmaple 33428 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le
`  K ) Y  <-> 
( M `  X
)  C_  ( M `  Y ) ) )
82, 3, 6pmaple 33428 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( le
`  K ) X  <-> 
( M `  Y
)  C_  ( M `  X ) ) )
983com23 1193 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y ( le
`  K ) X  <-> 
( M `  Y
)  C_  ( M `  X ) ) )
107, 9anbi12d 710 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X ( le `  K ) Y  /\  Y ( le `  K ) X )  <->  ( ( M `  X )  C_  ( M `  Y
)  /\  ( M `  Y )  C_  ( M `  X )
) ) )
115, 10bitr3d 255 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  <-> 
( ( M `  X )  C_  ( M `  Y )  /\  ( M `  Y
)  C_  ( M `  X ) ) ) )
12 eqss 3390 . 2  |-  ( ( M `  X )  =  ( M `  Y )  <->  ( ( M `  X )  C_  ( M `  Y
)  /\  ( M `  Y )  C_  ( M `  X )
) )
1311, 12syl6rbbr 264 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( M `  X )  =  ( M `  Y )  <-> 
X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3347   class class class wbr 4311   ` cfv 5437   Basecbs 14193   lecple 14264   Latclat 15234   HLchlt 33018   pmapcpmap 33164
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 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-poset 15135  df-plt 15147  df-lub 15163  df-glb 15164  df-join 15165  df-meet 15166  df-p0 15228  df-lat 15235  df-clat 15297  df-oposet 32844  df-ol 32846  df-oml 32847  df-covers 32934  df-ats 32935  df-atl 32966  df-cvlat 32990  df-hlat 33019  df-pmap 33171
This theorem is referenced by:  pmapeq0  33433  isline3  33443  lncvrelatN  33448
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