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Theorem tolat 25286
Description: A totally ordered set is a lattice. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
tolat  |-  TosetRel  C_  LatRel

Proof of Theorem tolat
Dummy variables  v  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 spwpr4c 14341 . . . . . . . . . . . 12  |-  ( ( x  e.  PosetRel  /\  u x v )  -> 
( x  sup w  { u ,  v } )  =  v )
21ancoms 439 . . . . . . . . . . 11  |-  ( ( u x v  /\  x  e.  PosetRel )  ->  (
x  sup w  { u ,  v } )  =  v )
3 posispre 25241 . . . . . . . . . . . . 13  |-  ( x  e.  PosetRel  ->  x  e. PresetRel )
4 eqid 2283 . . . . . . . . . . . . . 14  |-  dom  x  =  dom  x
54pre2befi2 25232 . . . . . . . . . . . . 13  |-  ( ( x  e. PresetRel  /\  u x v )  -> 
v  e.  dom  x
)
63, 5sylan 457 . . . . . . . . . . . 12  |-  ( ( x  e.  PosetRel  /\  u x v )  -> 
v  e.  dom  x
)
76ancoms 439 . . . . . . . . . . 11  |-  ( ( u x v  /\  x  e.  PosetRel )  ->  v  e.  dom  x )
82, 7eqeltrd 2357 . . . . . . . . . 10  |-  ( ( u x v  /\  x  e.  PosetRel )  ->  (
x  sup w  { u ,  v } )  e.  dom  x )
9 nfwpr4c 25285 . . . . . . . . . . . 12  |-  ( ( x  e.  PosetRel  /\  u x v )  -> 
( x  inf w  { u ,  v } )  =  u )
109ancoms 439 . . . . . . . . . . 11  |-  ( ( u x v  /\  x  e.  PosetRel )  ->  (
x  inf w  { u ,  v } )  =  u )
11 vex 2791 . . . . . . . . . . . . 13  |-  u  e. 
_V
12 vex 2791 . . . . . . . . . . . . 13  |-  v  e. 
_V
1311, 12breldm 4883 . . . . . . . . . . . 12  |-  ( u x v  ->  u  e.  dom  x )
1413adantr 451 . . . . . . . . . . 11  |-  ( ( u x v  /\  x  e.  PosetRel )  ->  u  e.  dom  x )
1510, 14eqeltrd 2357 . . . . . . . . . 10  |-  ( ( u x v  /\  x  e.  PosetRel )  ->  (
x  inf w  { u ,  v } )  e.  dom  x )
168, 15jca 518 . . . . . . . . 9  |-  ( ( u x v  /\  x  e.  PosetRel )  ->  (
( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  { u ,  v } )  e.  dom  x ) )
1716ex 423 . . . . . . . 8  |-  ( u x v  ->  (
x  e.  PosetRel  ->  (
( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  { u ,  v } )  e.  dom  x ) ) )
18 spwpr4c 14341 . . . . . . . . . . . . 13  |-  ( ( x  e.  PosetRel  /\  v
x u )  -> 
( x  sup w  { v ,  u } )  =  u )
1918ancoms 439 . . . . . . . . . . . 12  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  (
x  sup w  { v ,  u } )  =  u )
204pre2befi2 25232 . . . . . . . . . . . . . 14  |-  ( ( x  e. PresetRel  /\  v
x u )  ->  u  e.  dom  x )
213, 20sylan 457 . . . . . . . . . . . . 13  |-  ( ( x  e.  PosetRel  /\  v
x u )  ->  u  e.  dom  x )
2221ancoms 439 . . . . . . . . . . . 12  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  u  e.  dom  x )
2319, 22eqeltrd 2357 . . . . . . . . . . 11  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  (
x  sup w  { v ,  u } )  e.  dom  x )
24 prcom 3705 . . . . . . . . . . . 12  |-  { u ,  v }  =  { v ,  u }
25 oveq2 5866 . . . . . . . . . . . . 13  |-  ( { u ,  v }  =  { v ,  u }  ->  (
x  sup w  { u ,  v } )  =  ( x  sup w  { v ,  u } ) )
2625eleq1d 2349 . . . . . . . . . . . 12  |-  ( { u ,  v }  =  { v ,  u }  ->  (
( x  sup w  { u ,  v } )  e.  dom  x 
<->  ( x  sup w  { v ,  u } )  e.  dom  x ) )
2724, 26ax-mp 8 . . . . . . . . . . 11  |-  ( ( x  sup w  {
u ,  v } )  e.  dom  x  <->  ( x  sup w  {
v ,  u }
)  e.  dom  x
)
2823, 27sylibr 203 . . . . . . . . . 10  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  (
x  sup w  { u ,  v } )  e.  dom  x )
2924oveq2i 5869 . . . . . . . . . . 11  |-  ( x  inf w  { u ,  v } )  =  ( x  inf w  { v ,  u } )
30 nfwpr4c 25285 . . . . . . . . . . . . 13  |-  ( ( x  e.  PosetRel  /\  v
x u )  -> 
( x  inf w  { v ,  u } )  =  v )
3130ancoms 439 . . . . . . . . . . . 12  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  (
x  inf w  { v ,  u } )  =  v )
3212, 11breldm 4883 . . . . . . . . . . . . 13  |-  ( v x u  ->  v  e.  dom  x )
3332adantr 451 . . . . . . . . . . . 12  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  v  e.  dom  x )
3431, 33eqeltrd 2357 . . . . . . . . . . 11  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  (
x  inf w  { v ,  u } )  e.  dom  x )
3529, 34syl5eqel 2367 . . . . . . . . . 10  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  (
x  inf w  { u ,  v } )  e.  dom  x )
3628, 35jca 518 . . . . . . . . 9  |-  ( ( v x u  /\  x  e.  PosetRel )  ->  (
( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  { u ,  v } )  e.  dom  x ) )
3736ex 423 . . . . . . . 8  |-  ( v x u  ->  (
x  e.  PosetRel  ->  (
( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  { u ,  v } )  e.  dom  x ) ) )
3817, 37jaoi 368 . . . . . . 7  |-  ( ( u x v  \/  v x u )  ->  ( x  e.  PosetRel 
->  ( ( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  { u ,  v } )  e.  dom  x ) ) )
3938com12 27 . . . . . 6  |-  ( x  e.  PosetRel  ->  ( ( u x v  \/  v
x u )  -> 
( ( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  { u ,  v } )  e.  dom  x ) ) )
4039ralimdv 2622 . . . . 5  |-  ( x  e.  PosetRel  ->  ( A. v  e.  dom  x ( u x v  \/  v
x u )  ->  A. v  e.  dom  x ( ( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  {
u ,  v } )  e.  dom  x
) ) )
4140ralimdv 2622 . . . 4  |-  ( x  e.  PosetRel  ->  ( A. u  e.  dom  x A. v  e.  dom  x ( u x v  \/  v
x u )  ->  A. u  e.  dom  x A. v  e.  dom  x ( ( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  {
u ,  v } )  e.  dom  x
) ) )
4241imdistani 671 . . 3  |-  ( ( x  e.  PosetRel  /\  A. u  e.  dom  x A. v  e.  dom  x ( u x v  \/  v x u ) )  ->  ( x  e. 
PosetRel  /\  A. u  e. 
dom  x A. v  e.  dom  x ( ( x  sup w  {
u ,  v } )  e.  dom  x  /\  ( x  inf w  { u ,  v } )  e.  dom  x ) ) )
434istsr2 14327 . . 3  |-  ( x  e.  TosetRel 
<->  ( x  e.  PosetRel  /\  A. u  e.  dom  x A. v  e.  dom  x ( u x v  \/  v x u ) ) )
444isla 14342 . . 3  |-  ( x  e.  LatRel 
<->  ( x  e.  PosetRel  /\  A. u  e.  dom  x A. v  e.  dom  x ( ( x  sup w  { u ,  v } )  e.  dom  x  /\  ( x  inf w  {
u ,  v } )  e.  dom  x
) ) )
4542, 43, 443imtr4i 257 . 2  |-  ( x  e.  TosetRel  ->  x  e.  LatRel )
4645ssriv 3184 1  |-  TosetRel  C_  LatRel
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   {cpr 3641   class class class wbr 4023   dom cdm 4689  (class class class)co 5858   PosetRelcps 14301    TosetRel ctsr 14302    sup
w cspw 14303    inf
w cinf 14304   LatRelcla 14305  PresetRelcpresetrel 25215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-ps 14306  df-tsr 14307  df-spw 14308  df-nfw 14309  df-lar 14310  df-prs 25223
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