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Theorem lubsn 15926
Description: The least upper bound of a singleton. (chsupsn 26532 analog.) (Contributed by NM, 20-Oct-2011.)
Hypotheses
Ref Expression
lubsn.b  |-  B  =  ( Base `  K
)
lubsn.u  |-  U  =  ( lub `  K
)
Assertion
Ref Expression
lubsn  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( U `  { X } )  =  X )

Proof of Theorem lubsn
StepHypRef Expression
1 lubsn.u . . . 4  |-  U  =  ( lub `  K
)
2 eqid 2454 . . . 4  |-  ( join `  K )  =  (
join `  K )
3 simpl 455 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  K  e.  Lat )
4 simpr 459 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  e.  B )
51, 2, 3, 4, 4joinval 15837 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X ( join `  K ) X )  =  ( U `  { X ,  X }
) )
6 dfsn2 4029 . . . 4  |-  { X }  =  { X ,  X }
76fveq2i 5851 . . 3  |-  ( U `
 { X }
)  =  ( U `
 { X ,  X } )
85, 7syl6reqr 2514 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( U `  { X } )  =  ( X ( join `  K
) X ) )
9 lubsn.b . . 3  |-  B  =  ( Base `  K
)
109, 2latjidm 15906 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X ( join `  K ) X )  =  X )
118, 10eqtrd 2495 1  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( U `  { X } )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {csn 4016   {cpr 4018   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lubclub 15773   joincjn 15775   Latclat 15877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15759  df-poset 15777  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-lat 15878
This theorem is referenced by:  lubel  15954
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