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Theorem pmap11 29218
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 28820 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
2 pmap11.b . . . . 5  |-  B  =  ( Base `  K
)
3 eqid 2284 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
42, 3latasymb 14154 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X ( le `  K ) Y  /\  Y ( le `  K ) X )  <->  X  =  Y ) )
51, 4syl3an1 1217 . . 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 29217 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le
`  K ) Y  <-> 
( M `  X
)  C_  ( M `  Y ) ) )
82, 3, 6pmaple 29217 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( le
`  K ) X  <-> 
( M `  Y
)  C_  ( M `  X ) ) )
983com23 1159 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y ( le
`  K ) X  <-> 
( M `  Y
)  C_  ( M `  X ) ) )
107, 9anbi12d 693 . . 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 248 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  <-> 
( ( M `  X )  C_  ( M `  Y )  /\  ( M `  Y
)  C_  ( M `  X ) ) ) )
12 eqss 3195 . 2  |-  ( ( M `  X )  =  ( M `  Y )  <->  ( ( M `  X )  C_  ( M `  Y
)  /\  ( M `  Y )  C_  ( M `  X )
) )
1311, 12syl6rbbr 257 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( M `  X )  =  ( M `  Y )  <-> 
X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    C_ wss 3153   class class class wbr 4024   ` cfv 5221   Basecbs 13142   lecple 13209   Latclat 14145   HLchlt 28807   pmapcpmap 28953
This theorem is referenced by:  pmapeq0  29222  isline3  29232  lncvrelatN  29237
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-pmap 28960
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