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Theorem p0le 14392
Description: Poset zero (if defined) is less than any element. (Contributed by NM, 22-Oct-2011.)
Hypotheses
Ref Expression
p0le.b  |-  B  =  ( Base `  K
)
p0le.l  |-  .<_  =  ( le `  K )
p0le.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
p0le  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )

Proof of Theorem p0le
StepHypRef Expression
1 p0le.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2380 . . . 4  |-  ( glb `  K )  =  ( glb `  K )
3 p0le.z . . . 4  |-  .0.  =  ( 0. `  K )
41, 2, 3p0val 14390 . . 3  |-  ( K  e.  Poset  ->  .0.  =  ( ( glb `  K
) `  B )
)
543ad2ant1 978 . 2  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  =  ( ( glb `  K ) `  B
) )
6 simp1 957 . . 3  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  K  e.  Poset )
7 ssid 3303 . . . 4  |-  B  C_  B
87a1i 11 . . 3  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  B  C_  B )
94adantr 452 . . . . 5  |-  ( ( K  e.  Poset  /\  .0.  e.  B )  ->  .0.  =  ( ( glb `  K ) `  B
) )
10 simpr 448 . . . . 5  |-  ( ( K  e.  Poset  /\  .0.  e.  B )  ->  .0.  e.  B )
119, 10eqeltrrd 2455 . . . 4  |-  ( ( K  e.  Poset  /\  .0.  e.  B )  ->  (
( glb `  K
) `  B )  e.  B )
12113adant3 977 . . 3  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  (
( glb `  K
) `  B )  e.  B )
13 simp3 959 . . 3  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  X  e.  B )
14 p0le.l . . . 4  |-  .<_  =  ( le `  K )
151, 14, 2glble 14363 . . 3  |-  ( ( ( K  e.  Poset  /\  B  C_  B )  /\  ( ( ( glb `  K ) `  B
)  e.  B  /\  X  e.  B )
)  ->  ( ( glb `  K ) `  B )  .<_  X )
166, 8, 12, 13, 15syl22anc 1185 . 2  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  (
( glb `  K
) `  B )  .<_  X )
175, 16eqbrtrd 4166 1  |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3256   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   Posetcpo 14317   glbcglb 14320   0.cp0 14386
This theorem is referenced by:  op0le  29352  atl0le  29470
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-undef 6472  df-riota 6478  df-glb 14352  df-p0 14388
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