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Theorem lub0N 29924
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 3648 . . 3  |-  (/)  C_  ( Base `  K )
2 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2435 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
4 lub0.u . . . 4  |-  .1.  =  ( lub `  K )
52, 3, 4lubval 14428 . . 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 653 . 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 29919 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
9 ral0 3724 . . . . . . . 8  |-  A. y  e.  (/)  y ( le
`  K ) z
109a1bi 328 . . . . . . 7  |-  ( x ( le `  K
) z  <->  ( A. y  e.  (/)  y ( le `  K ) z  ->  x ( le `  K ) z ) )
1110ralbii 2721 . . . . . 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 3724 . . . . . . 7  |-  A. y  e.  (/)  y ( le
`  K ) x
1312biantrur 493 . . . . . 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 241 . . . . 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 4208 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x ( le `  K ) z  <->  x ( le `  K )  .0.  ) )
1615rspcv 3040 . . . . . . . 8  |-  (  .0. 
e.  ( Base `  K
)  ->  ( A. z  e.  ( Base `  K ) x ( le `  K ) z  ->  x ( le `  K )  .0.  ) )
17163ad2ant2 979 . . . . . . 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 29922 . . . . . . . 8  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( x ( le
`  K )  .0.  <->  x  =  .0.  ) )
19183adant2 976 . . . . . . 7  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( x
( le `  K
)  .0.  <->  x  =  .0.  ) )
2017, 19sylibd 206 . . . . . 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 29921 . . . . . . . . . . 11  |-  ( ( K  e.  OP  /\  z  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) z )
22213ad2antl1 1119 . . . . . . . . . 10  |-  ( ( ( K  e.  OP  /\  .0.  e.  ( Base `  K )  /\  x  e.  ( Base `  K
) )  /\  z  e.  ( Base `  K
) )  ->  .0.  ( le `  K ) z )
2322ex 424 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( z  e.  ( Base `  K
)  ->  .0.  ( le `  K ) z ) )
24 breq1 4207 . . . . . . . . . 10  |-  ( x  =  .0.  ->  (
x ( le `  K ) z  <->  .0.  ( le `  K ) z ) )
2524biimprcd 217 . . . . . . . . 9  |-  (  .0.  ( le `  K
) z  ->  (
x  =  .0.  ->  x ( le `  K
) z ) )
2623, 25syl6 31 . . . . . . . 8  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( z  e.  ( Base `  K
)  ->  ( x  =  .0.  ->  x ( le `  K ) z ) ) )
2726com23 74 . . . . . . 7  |-  ( ( K  e.  OP  /\  .0.  e.  ( Base `  K
)  /\  x  e.  ( Base `  K )
)  ->  ( x  =  .0.  ->  ( z  e.  ( Base `  K
)  ->  x ( le `  K ) z ) ) )
2827ralrimdv 2787 . . . . . 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 184 . . . . 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 251 . . . 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 6568 . . 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 650 . 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 2467 1  |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   (/)c0 3620   class class class wbr 4204   ` cfv 5446   iota_crio 6534   Basecbs 13461   lecple 13528   lubclub 14391   0.cp0 14458   OPcops 29907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-undef 6535  df-riota 6541  df-poset 14395  df-lub 14423  df-glb 14424  df-p0 14460  df-oposet 29911
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