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Theorem pmap11 30398
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 30000 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
2 pmap11.b . . . . 5  |-  B  =  ( Base `  K
)
3 eqid 2435 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
42, 3latasymb 14471 . . . 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 30397 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le
`  K ) Y  <-> 
( M `  X
)  C_  ( M `  Y ) ) )
82, 3, 6pmaple 30397 . . . . 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 692 . . 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 247 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  <-> 
( ( M `  X )  C_  ( M `  Y )  /\  ( M `  Y
)  C_  ( M `  X ) ) ) )
12 eqss 3355 . 2  |-  ( ( M `  X )  =  ( M `  Y )  <->  ( ( M `  X )  C_  ( M `  Y
)  /\  ( M `  Y )  C_  ( M `  X )
) )
1311, 12syl6rbbr 256 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 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   class class class wbr 4204   ` cfv 5445   Basecbs 13457   lecple 13524   Latclat 14462   HLchlt 29987   pmapcpmap 30133
This theorem is referenced by:  pmapeq0  30402  isline3  30412  lncvrelatN  30417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-pmap 30140
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