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Theorem mrelatglb 14603
Description: Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypotheses
Ref Expression
mreclat.i  |-  I  =  (toInc `  C )
mrelatglb.g  |-  G  =  ( glb `  I
)
Assertion
Ref Expression
mrelatglb  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  ( G `
 U )  = 
|^| U )

Proof of Theorem mrelatglb
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 14574 . . 3  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
433ad2ant1 978 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  C  =  ( Base `  I
) )
5 mrelatglb.g . . 3  |-  G  =  ( glb `  I
)
65a1i 11 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  G  =  ( glb `  I
) )
72ipopos 14579 . . 3  |-  I  e. 
Poset
87a1i 11 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  I  e. 
Poset )
9 simp2 958 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  U  C_  C )
10 mreintcl 13813 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  |^| U  e.  C )
11 intss1 4058 . . . 4  |-  ( x  e.  U  ->  |^| U  C_  x )
1211adantl 453 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  |^| U  C_  x )
13 simpl1 960 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
1410adantr 452 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  |^| U  e.  C )
159sselda 3341 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  x  e.  C )
162, 1ipole 14577 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  |^| U  e.  C  /\  x  e.  C )  ->  ( |^| U ( le `  I ) x  <->  |^| U  C_  x
) )
1713, 14, 15, 16syl3anc 1184 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  ( |^| U ( le `  I ) x  <->  |^| U  C_  x ) )
1812, 17mpbird 224 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  |^| U
( le `  I
) x )
19 simpll1 996 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
20 simplr 732 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  y  e.  C )
21 simpl2 961 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  ->  U  C_  C )
2221sselda 3341 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  x  e.  C )
232, 1ipole 14577 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  y  e.  C  /\  x  e.  C )  ->  (
y ( le `  I ) x  <->  y  C_  x ) )
2419, 20, 22, 23syl3anc 1184 . . . . . . 7  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  (
y ( le `  I ) x  <->  y  C_  x ) )
2524biimpd 199 . . . . . 6  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  (
y ( le `  I ) x  -> 
y  C_  x )
)
2625ralimdva 2777 . . . . 5  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  ->  ( A. x  e.  U  y ( le `  I ) x  ->  A. x  e.  U  y  C_  x ) )
27263impia 1150 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C  /\  A. x  e.  U  y ( le `  I ) x )  ->  A. x  e.  U  y  C_  x )
28 ssint 4059 . . . 4  |-  ( y 
C_  |^| U  <->  A. x  e.  U  y  C_  x )
2927, 28sylibr 204 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C  /\  A. x  e.  U  y ( le `  I ) x )  ->  y  C_  |^| U )
30 simp11 987 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C  /\  A. x  e.  U  y ( le `  I ) x )  ->  C  e.  (Moore `  X ) )
31 simp2 958 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C  /\  A. x  e.  U  y ( le `  I ) x )  ->  y  e.  C )
32103ad2ant1 978 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C  /\  A. x  e.  U  y ( le `  I ) x )  ->  |^| U  e.  C )
332, 1ipole 14577 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  y  e.  C  /\  |^| U  e.  C )  ->  (
y ( le `  I ) |^| U  <->  y 
C_  |^| U ) )
3430, 31, 32, 33syl3anc 1184 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C  /\  A. x  e.  U  y ( le `  I ) x )  ->  ( y
( le `  I
) |^| U  <->  y  C_  |^| U ) )
3529, 34mpbird 224 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C  /\  A. x  e.  U  y ( le `  I ) x )  ->  y ( le `  I ) |^| U )
361, 4, 6, 8, 9, 10, 18, 35posglbd 14569 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  ( G `
 U )  = 
|^| U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2698    C_ wss 3313   (/)c0 3621   |^|cint 4043   class class class wbr 4205   ` cfv 5447   Basecbs 13462   lecple 13529  Moorecmre 13800   Posetcpo 14390   glbcglb 14393  toInccipo 14570
This theorem is referenced by:  mreclat  14606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-uz 10482  df-fz 11037  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-tset 13541  df-ple 13542  df-ocomp 13543  df-mre 13804  df-poset 14396  df-lub 14424  df-glb 14425  df-odu 14549  df-ipo 14571
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