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Theorem p0le 15813
Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)
Hypotheses
Ref Expression
p0le.b  |-  B  =  ( Base `  K
)
p0le.g  |-  G  =  ( glb `  K
)
p0le.l  |-  .<_  =  ( le `  K )
p0le.0  |-  .0.  =  ( 0. `  K )
p0le.k  |-  ( ph  ->  K  e.  V )
p0le.x  |-  ( ph  ->  X  e.  B )
p0le.d  |-  ( ph  ->  B  e.  dom  G
)
Assertion
Ref Expression
p0le  |-  ( ph  ->  .0.  .<_  X )

Proof of Theorem p0le
StepHypRef Expression
1 p0le.k . . 3  |-  ( ph  ->  K  e.  V )
2 p0le.b . . . 4  |-  B  =  ( Base `  K
)
3 p0le.g . . . 4  |-  G  =  ( glb `  K
)
4 p0le.0 . . . 4  |-  .0.  =  ( 0. `  K )
52, 3, 4p0val 15811 . . 3  |-  ( K  e.  V  ->  .0.  =  ( G `  B ) )
61, 5syl 16 . 2  |-  ( ph  ->  .0.  =  ( G `
 B ) )
7 p0le.l . . 3  |-  .<_  =  ( le `  K )
8 p0le.d . . 3  |-  ( ph  ->  B  e.  dom  G
)
9 p0le.x . . 3  |-  ( ph  ->  X  e.  B )
102, 7, 3, 1, 8, 9glble 15770 . 2  |-  ( ph  ->  ( G `  B
)  .<_  X )
116, 10eqbrtrd 4404 1  |-  ( ph  ->  .0.  .<_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1836   class class class wbr 4384   dom cdm 4930   ` cfv 5513   Basecbs 14657   lecple 14732   glbcglb 15712   0.cp0 15807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-glb 15745  df-p0 15809
This theorem is referenced by:  op0le  35363  atl0le  35481
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