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Theorem clatl 15620
Description: A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5108 to shorten proof and eliminate joindmss 15511 and meetdmss 15525?
Assertion
Ref Expression
clatl  |-  ( K  e.  CLat  ->  K  e. 
Lat )

Proof of Theorem clatl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2467 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
3 simpl 457 . . . . . . 7  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  ->  K  e.  Poset )
41, 2, 3joindmss 15511 . . . . . 6  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  ->  dom  ( join `  K
)  C_  ( ( Base `  K )  X.  ( Base `  K
) ) )
5 relxp 5116 . . . . . . . 8  |-  Rel  (
( Base `  K )  X.  ( Base `  K
) )
65a1i 11 . . . . . . 7  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  ->  Rel  ( ( Base `  K
)  X.  ( Base `  K ) ) )
7 opelxp 5035 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( ( Base `  K
)  X.  ( Base `  K ) )  <->  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )
8 vex 3121 . . . . . . . . . . . . 13  |-  x  e. 
_V
9 vex 3121 . . . . . . . . . . . . 13  |-  y  e. 
_V
108, 9prss 4187 . . . . . . . . . . . 12  |-  ( ( x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  <->  { x ,  y }  C_  ( Base `  K )
)
117, 10sylbb 197 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( ( Base `  K
)  X.  ( Base `  K ) )  ->  { x ,  y }  C_  ( Base `  K ) )
12 prex 4695 . . . . . . . . . . . 12  |-  { x ,  y }  e.  _V
1312elpw 4022 . . . . . . . . . . 11  |-  ( { x ,  y }  e.  ~P ( Base `  K )  <->  { x ,  y }  C_  ( Base `  K )
)
1411, 13sylibr 212 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( ( Base `  K
)  X.  ( Base `  K ) )  ->  { x ,  y }  e.  ~P ( Base `  K ) )
15 eleq2 2540 . . . . . . . . . 10  |-  ( dom  ( lub `  K
)  =  ~P ( Base `  K )  -> 
( { x ,  y }  e.  dom  ( lub `  K )  <->  { x ,  y }  e.  ~P ( Base `  K ) ) )
1614, 15syl5ibr 221 . . . . . . . . 9  |-  ( dom  ( lub `  K
)  =  ~P ( Base `  K )  -> 
( <. x ,  y
>.  e.  ( ( Base `  K )  X.  ( Base `  K ) )  ->  { x ,  y }  e.  dom  ( lub `  K ) ) )
1716adantl 466 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  -> 
( <. x ,  y
>.  e.  ( ( Base `  K )  X.  ( Base `  K ) )  ->  { x ,  y }  e.  dom  ( lub `  K ) ) )
18 eqid 2467 . . . . . . . . 9  |-  ( lub `  K )  =  ( lub `  K )
198a1i 11 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  ->  x  e.  _V )
209a1i 11 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  -> 
y  e.  _V )
2118, 2, 3, 19, 20joindef 15508 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  -> 
( <. x ,  y
>.  e.  dom  ( join `  K )  <->  { x ,  y }  e.  dom  ( lub `  K
) ) )
2217, 21sylibrd 234 . . . . . . 7  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  -> 
( <. x ,  y
>.  e.  ( ( Base `  K )  X.  ( Base `  K ) )  ->  <. x ,  y
>.  e.  dom  ( join `  K ) ) )
236, 22relssdv 5101 . . . . . 6  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  -> 
( ( Base `  K
)  X.  ( Base `  K ) )  C_  dom  ( join `  K
) )
244, 23eqssd 3526 . . . . 5  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  ->  dom  ( join `  K
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
2524ex 434 . . . 4  |-  ( K  e.  Poset  ->  ( dom  ( lub `  K )  =  ~P ( Base `  K )  ->  dom  ( join `  K )  =  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
26 eqid 2467 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
27 simpl 457 . . . . . . 7  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  ->  K  e.  Poset )
281, 26, 27meetdmss 15525 . . . . . 6  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  ->  dom  ( meet `  K
)  C_  ( ( Base `  K )  X.  ( Base `  K
) ) )
295a1i 11 . . . . . . 7  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  ->  Rel  ( ( Base `  K
)  X.  ( Base `  K ) ) )
30 eleq2 2540 . . . . . . . . . 10  |-  ( dom  ( glb `  K
)  =  ~P ( Base `  K )  -> 
( { x ,  y }  e.  dom  ( glb `  K )  <->  { x ,  y }  e.  ~P ( Base `  K ) ) )
3114, 30syl5ibr 221 . . . . . . . . 9  |-  ( dom  ( glb `  K
)  =  ~P ( Base `  K )  -> 
( <. x ,  y
>.  e.  ( ( Base `  K )  X.  ( Base `  K ) )  ->  { x ,  y }  e.  dom  ( glb `  K ) ) )
3231adantl 466 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  -> 
( <. x ,  y
>.  e.  ( ( Base `  K )  X.  ( Base `  K ) )  ->  { x ,  y }  e.  dom  ( glb `  K ) ) )
33 eqid 2467 . . . . . . . . 9  |-  ( glb `  K )  =  ( glb `  K )
348a1i 11 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  ->  x  e.  _V )
359a1i 11 . . . . . . . . 9  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  -> 
y  e.  _V )
3633, 26, 27, 34, 35meetdef 15522 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  -> 
( <. x ,  y
>.  e.  dom  ( meet `  K )  <->  { x ,  y }  e.  dom  ( glb `  K
) ) )
3732, 36sylibrd 234 . . . . . . 7  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  -> 
( <. x ,  y
>.  e.  ( ( Base `  K )  X.  ( Base `  K ) )  ->  <. x ,  y
>.  e.  dom  ( meet `  K ) ) )
3829, 37relssdv 5101 . . . . . 6  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  -> 
( ( Base `  K
)  X.  ( Base `  K ) )  C_  dom  ( meet `  K
) )
3928, 38eqssd 3526 . . . . 5  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  ->  dom  ( meet `  K
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
4039ex 434 . . . 4  |-  ( K  e.  Poset  ->  ( dom  ( glb `  K )  =  ~P ( Base `  K )  ->  dom  ( meet `  K )  =  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
4125, 40anim12d 563 . . 3  |-  ( K  e.  Poset  ->  ( ( dom  ( lub `  K
)  =  ~P ( Base `  K )  /\  dom  ( glb `  K
)  =  ~P ( Base `  K ) )  ->  ( dom  ( join `  K )  =  ( ( Base `  K
)  X.  ( Base `  K ) )  /\  dom  ( meet `  K
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) ) ) )
4241imdistani 690 . 2  |-  ( ( K  e.  Poset  /\  ( dom  ( lub `  K
)  =  ~P ( Base `  K )  /\  dom  ( glb `  K
)  =  ~P ( Base `  K ) ) )  ->  ( K  e.  Poset  /\  ( dom  ( join `  K )  =  ( ( Base `  K )  X.  ( Base `  K ) )  /\  dom  ( meet `  K )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
431, 18, 33isclat 15613 . 2  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  ( lub `  K )  =  ~P ( Base `  K
)  /\  dom  ( glb `  K )  =  ~P ( Base `  K )
) ) )
441, 2, 26islat 15551 . 2  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  ( join `  K )  =  ( ( Base `  K )  X.  ( Base `  K ) )  /\  dom  ( meet `  K )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
4542, 43, 443imtr4i 266 1  |-  ( K  e.  CLat  ->  K  e. 
Lat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   {cpr 4035   <.cop 4039    X. cxp 5003   dom cdm 5005   Rel wrel 5010   ` cfv 5594   Basecbs 14507   Posetcpo 15444   lubclub 15446   glbcglb 15447   joincjn 15448   meetcmee 15449   Latclat 15549   CLatccla 15611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-oprab 6299  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-lat 15550  df-clat 15612
This theorem is referenced by:  lubel  15626  lubun  15627  clatleglb  15630
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