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Theorem pmap11 34433
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 34035 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
2 pmap11.b . . . . 5  |-  B  =  ( Base `  K
)
3 eqid 2460 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
42, 3latasymb 15530 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X ( le `  K ) Y  /\  Y ( le `  K ) X )  <->  X  =  Y ) )
51, 4syl3an1 1256 . . 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 34432 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le
`  K ) Y  <-> 
( M `  X
)  C_  ( M `  Y ) ) )
82, 3, 6pmaple 34432 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( le
`  K ) X  <-> 
( M `  Y
)  C_  ( M `  X ) ) )
983com23 1197 . . . 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 3512 . 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 968    = wceq 1374    e. wcel 1762    C_ wss 3469   class class class wbr 4440   ` cfv 5579   Basecbs 14479   lecple 14551   Latclat 15521   HLchlt 34022   pmapcpmap 34168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-pmap 34175
This theorem is referenced by:  pmapeq0  34437  isline3  34447  lncvrelatN  34452
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