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Theorem mrelatglb 14531
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 2381 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 14502 . . 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 14507 . . 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 13741 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  |^| U  e.  C )
11 intss1 4001 . . . 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 3285 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  x  e.  C )
162, 1ipole 14505 . . . 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 3285 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  x  e.  C )
232, 1ipole 14505 . . . . . . . 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 2721 . . . . 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 4002 . . . 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 14505 . . . 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 14497 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 1717    =/= wne 2544   A.wral 2643    C_ wss 3257   (/)c0 3565   |^|cint 3986   class class class wbr 4147   ` cfv 5388   Basecbs 13390   lecple 13457  Moorecmre 13728   Posetcpo 14318   glbcglb 14321  toInccipo 14498
This theorem is referenced by:  mreclat  14534
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-er 6835  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-nn 9927  df-2 9984  df-3 9985  df-4 9986  df-5 9987  df-6 9988  df-7 9989  df-8 9990  df-9 9991  df-10 9992  df-n0 10148  df-z 10209  df-dec 10309  df-uz 10415  df-fz 10970  df-struct 13392  df-ndx 13393  df-slot 13394  df-base 13395  df-sets 13396  df-tset 13469  df-ple 13470  df-ocomp 13471  df-mre 13732  df-poset 14324  df-lub 14352  df-glb 14353  df-odu 14477  df-ipo 14499
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