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Theorem pmap11 29118
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 28720 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
2 pmap11.b . . . . 5  |-  B  =  ( Base `  K
)
3 eqid 2258 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
42, 3latasymb 14122 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X ( le `  K ) Y  /\  Y ( le `  K ) X )  <->  X  =  Y ) )
51, 4syl3an1 1220 . . 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 29117 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( le
`  K ) Y  <-> 
( M `  X
)  C_  ( M `  Y ) ) )
82, 3, 6pmaple 29117 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( le
`  K ) X  <-> 
( M `  Y
)  C_  ( M `  X ) ) )
983com23 1162 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y ( le
`  K ) X  <-> 
( M `  Y
)  C_  ( M `  X ) ) )
107, 9anbi12d 694 . . 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 3169 . 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 939    = wceq 1619    e. wcel 1621    C_ wss 3127   class class class wbr 3997   ` cfv 4673   Basecbs 13110   lecple 13177   Latclat 14113   HLchlt 28707   pmapcpmap 28853
This theorem is referenced by:  pmapeq0  29122  isline3  29132  lncvrelatN  29137
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-pmap 28860
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