MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clatl Structured version   Unicode version

Theorem clatl 15307
Description: A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 4960 to shorten proof and eliminate joindmss 15198 and meetdmss 15212?
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 2443 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2443 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
3 simpl 457 . . . . . . 7  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  ->  K  e.  Poset )
41, 2, 3joindmss 15198 . . . . . 6  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  ->  dom  ( join `  K
)  C_  ( ( Base `  K )  X.  ( Base `  K
) ) )
5 relxp 4968 . . . . . . . 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 4890 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( ( Base `  K
)  X.  ( Base `  K ) )  <->  ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
) )
8 vex 2996 . . . . . . . . . . . . 13  |-  x  e. 
_V
9 vex 2996 . . . . . . . . . . . . 13  |-  y  e. 
_V
108, 9prss 4048 . . . . . . . . . . . 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 4555 . . . . . . . . . . . 12  |-  { x ,  y }  e.  _V
1312elpw 3887 . . . . . . . . . . 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 2504 . . . . . . . . . 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 2443 . . . . . . . . 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 15195 . . . . . . . 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 4953 . . . . . 6  |-  ( ( K  e.  Poset  /\  dom  ( lub `  K )  =  ~P ( Base `  K ) )  -> 
( ( Base `  K
)  X.  ( Base `  K ) )  C_  dom  ( join `  K
) )
244, 23eqssd 3394 . . . . 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 2443 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
27 simpl 457 . . . . . . 7  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  ->  K  e.  Poset )
281, 26, 27meetdmss 15212 . . . . . 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 2504 . . . . . . . . . 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 2443 . . . . . . . . 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 15209 . . . . . . . 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 4953 . . . . . 6  |-  ( ( K  e.  Poset  /\  dom  ( glb `  K )  =  ~P ( Base `  K ) )  -> 
( ( Base `  K
)  X.  ( Base `  K ) )  C_  dom  ( meet `  K
) )
3928, 38eqssd 3394 . . . . 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 15300 . 2  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  ( lub `  K )  =  ~P ( Base `  K
)  /\  dom  ( glb `  K )  =  ~P ( Base `  K )
) ) )
441, 2, 26islat 15238 . 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 1369    e. wcel 1756   _Vcvv 2993    C_ wss 3349   ~Pcpw 3881   {cpr 3900   <.cop 3904    X. cxp 4859   dom cdm 4861   Rel wrel 4866   ` cfv 5439   Basecbs 14195   Posetcpo 15131   lubclub 15133   glbcglb 15134   joincjn 15135   meetcmee 15136   Latclat 15236   CLatccla 15298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-oprab 6116  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-lat 15237  df-clat 15299
This theorem is referenced by:  lubel  15313  lubun  15314  clatleglb  15317
  Copyright terms: Public domain W3C validator