Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lub0N Unicode version

Theorem lub0N 29379
Description: The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lub0.u  |-  .1.  =  ( lub `  K )
lub0.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
lub0N  |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )

Proof of Theorem lub0N
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3483 . . 3  |-  (/)  C_  ( Base `  K )
2 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
4 lub0.u . . . 4  |-  .1.  =  ( lub `  K )
52, 3, 4lubval 14113 . . 3  |-  ( ( K  e.  OP  /\  (/)  C_  ( Base `  K
) )  ->  (  .1.  `  (/) )  =  (
iota_ x  e.  ( Base `  K ) ( A. y  e.  (/)  y ( le `  K ) x  /\  A. z  e.  ( Base `  K ) ( A. y  e.  (/)  y ( le `  K ) z  ->  x ( le `  K ) z ) ) ) )
61, 5mpan2 652 . 2  |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  (
iota_ x  e.  ( Base `  K ) ( A. y  e.  (/)  y ( le `  K ) x  /\  A. z  e.  ( Base `  K ) ( A. y  e.  (/)  y ( le `  K ) z  ->  x ( le `  K ) z ) ) ) )
7 lub0.z . . . 4  |-  .0.  =  ( 0. `  K )
82, 7op0cl 29374 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
9 ral0 3558 . . . . . . . 8  |-  A. y  e.  (/)  y ( le
`  K ) z
109a1bi 327 . . . . . . 7  |-  ( x ( le `  K
) z  <->  ( A. y  e.  (/)  y ( le `  K ) z  ->  x ( le `  K ) z ) )
1110ralbii 2567 . . . . . 6  |-  ( A. z  e.  ( Base `  K ) x ( le `  K ) z  <->  A. z  e.  (
Base `  K )
( A. y  e.  (/)  y ( le `  K ) z  ->  x ( le `  K ) z ) )
12 ral0 3558 . . . . . . 7  |-  A. y  e.  (/)  y ( le
`  K ) x
1312biantrur 492 . . . . . 6  |-  ( A. z  e.  ( Base `  K ) ( A. y  e.  (/)  y ( le `  K ) z  ->  x ( le `  K ) z )  <->  ( A. y  e.  (/)  y ( le
`  K ) x  /\  A. z  e.  ( Base `  K
) ( A. y  e.  (/)  y ( le
`  K ) z  ->  x ( le
`  K ) z ) ) )
1411, 13bitri 240 . . . . 5  |-  ( A. z  e.  ( Base `  K ) x ( le `  K ) z  <->  ( A. y  e.  (/)  y ( le
`  K ) x  /\  A. z  e.  ( Base `  K
) ( A. y  e.  (/)  y ( le
`  K ) z  ->  x ( le
`  K ) z ) ) )
15 breq2 4027 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x ( le `  K ) z  <->  x ( le `  K )  .0.  ) )
1615rspcv 2880 . . . . . . . 8  |-  (  .0. 
e.  ( Base `  K
)  ->  ( A. z  e.  ( Base `  K ) x ( le `  K ) z  ->  x ( le `  K )  .0.  ) )
17163ad2ant2 977 . . . . . . 7  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( A. z  e.  ( Base `  K ) x ( le `  K ) z  ->  x ( le `  K )  .0.  ) )
182, 3, 7ople0 29377 . . . . . . . 8  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( x ( le
`  K )  .0.  <->  x  =  .0.  ) )
19183adant2 974 . . . . . . 7  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( x
( le `  K
)  .0.  <->  x  =  .0.  ) )
2017, 19sylibd 205 . . . . . 6  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( A. z  e.  ( Base `  K ) x ( le `  K ) z  ->  x  =  .0.  ) )
212, 3, 7op0le 29376 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  z  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) z )
22213ad2antl1 1117 . . . . . . . . . 10  |-  ( ( ( K  e.  OP  /\  .0.  e.  ( Base `  K )  /\  x  e.  ( Base `  K
) )  /\  z  e.  ( Base `  K
) )  ->  .0.  ( le `  K ) z )
2322ex 423 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( z  e.  ( Base `  K
)  ->  .0.  ( le `  K ) z ) )
24 breq1 4026 . . . . . . . . . 10  |-  ( x  =  .0.  ->  (
x ( le `  K ) z  <->  .0.  ( le `  K ) z ) )
2524biimprcd 216 . . . . . . . . 9  |-  (  .0.  ( le `  K
) z  ->  (
x  =  .0.  ->  x ( le `  K
) z ) )
2623, 25syl6 29 . . . . . . . 8  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( z  e.  ( Base `  K
)  ->  ( x  =  .0.  ->  x ( le `  K ) z ) ) )
2726com23 72 . . . . . . 7  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( x  =  .0.  ->  ( z  e.  ( Base `  K
)  ->  x ( le `  K ) z ) ) )
2827ralrimdv 2632 . . . . . 6  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( x  =  .0.  ->  A. z  e.  ( Base `  K
) x ( le
`  K ) z ) )
2920, 28impbid 183 . . . . 5  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( A. z  e.  ( Base `  K ) x ( le `  K ) z  <->  x  =  .0.  ) )
3014, 29syl5bbr 250 . . . 4  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( ( A. y  e.  (/)  y ( le `  K ) x  /\  A. z  e.  ( Base `  K
) ( A. y  e.  (/)  y ( le
`  K ) z  ->  x ( le
`  K ) z ) )  <->  x  =  .0.  ) )
3130riota5OLD 6331 . . 3  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
) )  ->  ( iota_ x  e.  ( Base `  K ) ( A. y  e.  (/)  y ( le `  K ) x  /\  A. z  e.  ( Base `  K
) ( A. y  e.  (/)  y ( le
`  K ) z  ->  x ( le
`  K ) z ) ) )  =  .0.  )
328, 31mpdan 649 . 2  |-  ( K  e.  OP  ->  ( iota_ x  e.  ( Base `  K ) ( A. y  e.  (/)  y ( le `  K ) x  /\  A. z  e.  ( Base `  K
) ( A. y  e.  (/)  y ( le
`  K ) z  ->  x ( le
`  K ) z ) ) )  =  .0.  )
336, 32eqtrd 2315 1  |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ` cfv 5255   iota_crio 6297   Basecbs 13148   lecple 13215   lubclub 14076   0.cp0 14143   OPcops 29362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-glb 14109  df-p0 14145  df-oposet 29366
  Copyright terms: Public domain W3C validator