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Theorem mrelatglb 14303
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 2296 . 2  |-  ( le
`  I )  =  ( le `  I
)
2 mreclat.i . . . 4  |-  I  =  (toInc `  C )
32ipobas 14274 . . 3  |-  ( C  e.  (Moore `  X
)  ->  C  =  ( Base `  I )
)
433ad2ant1 976 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  C  =  ( Base `  I
) )
5 mrelatglb.g . . 3  |-  G  =  ( glb `  I
)
65a1i 10 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  G  =  ( glb `  I
) )
72ipopos 14279 . . 3  |-  I  e. 
Poset
87a1i 10 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  I  e. 
Poset )
9 simp2 956 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  U  C_  C )
10 mreintcl 13513 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  |^| U  e.  C )
11 intss1 3893 . . . 4  |-  ( x  e.  U  ->  |^| U  C_  x )
1211adantl 452 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  |^| U  C_  x )
13 simpl1 958 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
1410adantr 451 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  |^| U  e.  C )
159sselda 3193 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  x  e.  C )
162, 1ipole 14277 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  |^| U  e.  C  /\  x  e.  C )  ->  ( |^| U ( le `  I ) x  <->  |^| U  C_  x
) )
1713, 14, 15, 16syl3anc 1182 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  ( |^| U ( le `  I ) x  <->  |^| U  C_  x ) )
1812, 17mpbird 223 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  x  e.  U )  ->  |^| U
( le `  I
) x )
19 simpll1 994 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  C  e.  (Moore `  X )
)
20 simplr 731 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  y  e.  C )
21 simpl2 959 . . . . . . . . 9  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  ->  U  C_  C )
2221sselda 3193 . . . . . . . 8  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  x  e.  C )
232, 1ipole 14277 . . . . . . . 8  |-  ( ( C  e.  (Moore `  X )  /\  y  e.  C  /\  x  e.  C )  ->  (
y ( le `  I ) x  <->  y  C_  x ) )
2419, 20, 22, 23syl3anc 1182 . . . . . . 7  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  (
y ( le `  I ) x  <->  y  C_  x ) )
2524biimpd 198 . . . . . 6  |-  ( ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C )  /\  x  e.  U )  ->  (
y ( le `  I ) x  -> 
y  C_  x )
)
2625ralimdva 2634 . . . . 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 1148 . . . 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 3894 . . . 4  |-  ( y 
C_  |^| U  <->  A. x  e.  U  y  C_  x )
2927, 28sylibr 203 . . 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 985 . . . 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 956 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  /\  y  e.  C  /\  A. x  e.  U  y ( le `  I ) x )  ->  y  e.  C )
32103ad2ant1 976 . . . 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 14277 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  y  e.  C  /\  |^| U  e.  C )  ->  (
y ( le `  I ) |^| U  <->  y 
C_  |^| U ) )
3430, 31, 32, 33syl3anc 1182 . . 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 223 . 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 14269 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  ( G `
 U )  = 
|^| U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   |^|cint 3878   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231  Moorecmre 13500   Posetcpo 14090   glbcglb 14093  toInccipo 14270
This theorem is referenced by:  mreclat  14306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-tset 13243  df-ple 13244  df-ocomp 13245  df-mre 13504  df-poset 14096  df-lub 14124  df-glb 14125  df-odu 14249  df-ipo 14271
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