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Theorem mrelatglb 14565
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 2404 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 14536 . . 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 14541 . . 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 13775 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  |^| U  e.  C )
11 intss1 4025 . . . 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 3308 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  x  e.  C )
162, 1ipole 14539 . . . 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 3308 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  x  e.  C )
232, 1ipole 14539 . . . . . . . 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 2744 . . . . 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 4026 . . . 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 14539 . . . 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 14531 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 1649    e. wcel 1721    =/= wne 2567   A.wral 2666    C_ wss 3280   (/)c0 3588   |^|cint 4010   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491  Moorecmre 13762   Posetcpo 14352   glbcglb 14355  toInccipo 14532
This theorem is referenced by:  mreclat  14568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-tset 13503  df-ple 13504  df-ocomp 13505  df-mre 13766  df-poset 14358  df-lub 14386  df-glb 14387  df-odu 14511  df-ipo 14533
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