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Theorem lub0N 32522
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 eqid 2441 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2441 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 lub0.u . . 3  |-  .1.  =  ( lub `  K )
4 biid 236 . . 3  |-  ( ( A. y  e.  (/)  y ( le `  K ) x  /\  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 ) ) )
5 id 22 . . 3  |-  ( K  e.  OP  ->  K  e.  OP )
6 0ss 3663 . . . 4  |-  (/)  C_  ( Base `  K )
76a1i 11 . . 3  |-  ( K  e.  OP  ->  (/)  C_  ( Base `  K ) )
81, 2, 3, 4, 5, 7lubval 15150 . 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 ) ) ) )
9 lub0.z . . . 4  |-  .0.  =  ( 0. `  K )
101, 9op0cl 32517 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
11 ral0 3781 . . . . . . 7  |-  A. y  e.  (/)  y ( le
`  K ) z
1211a1bi 337 . . . . . 6  |-  ( x ( le `  K
) z  <->  ( A. y  e.  (/)  y ( le `  K ) z  ->  x ( le `  K ) z ) )
1312ralbii 2737 . . . . 5  |-  ( 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 ) )
14 ral0 3781 . . . . . 6  |-  A. y  e.  (/)  y ( le
`  K ) x
1514biantrur 503 . . . . 5  |-  ( 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 ) ) )
1613, 15bitri 249 . . . 4  |-  ( 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 ) ) )
1710adantr 462 . . . . . . 7  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  ->  .0.  e.  ( Base `  K
) )
18 breq2 4293 . . . . . . . 8  |-  ( z  =  .0.  ->  (
x ( le `  K ) z  <->  x ( le `  K )  .0.  ) )
1918rspcv 3066 . . . . . . 7  |-  (  .0. 
e.  ( Base `  K
)  ->  ( A. z  e.  ( Base `  K ) x ( le `  K ) z  ->  x ( le `  K )  .0.  ) )
2017, 19syl 16 . . . . . 6  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( A. z  e.  ( Base `  K
) x ( le
`  K ) z  ->  x ( le
`  K )  .0.  ) )
211, 2, 9ople0 32520 . . . . . 6  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( x ( le
`  K )  .0.  <->  x  =  .0.  ) )
2220, 21sylibd 214 . . . . 5  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( A. z  e.  ( Base `  K
) x ( le
`  K ) z  ->  x  =  .0.  ) )
231, 2, 9op0le 32519 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  z  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) z )
2423adantlr 709 . . . . . . . . 9  |-  ( ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  /\  z  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) z )
2524ex 434 . . . . . . . 8  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( z  e.  (
Base `  K )  ->  .0.  ( le `  K ) z ) )
26 breq1 4292 . . . . . . . . 9  |-  ( x  =  .0.  ->  (
x ( le `  K ) z  <->  .0.  ( le `  K ) z ) )
2726biimprcd 225 . . . . . . . 8  |-  (  .0.  ( le `  K
) z  ->  (
x  =  .0.  ->  x ( le `  K
) z ) )
2825, 27syl6 33 . . . . . . 7  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( z  e.  (
Base `  K )  ->  ( x  =  .0. 
->  x ( le `  K ) z ) ) )
2928com23 78 . . . . . 6  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( x  =  .0. 
->  ( z  e.  (
Base `  K )  ->  x ( le `  K ) z ) ) )
3029ralrimdv 2803 . . . . 5  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( x  =  .0. 
->  A. z  e.  (
Base `  K )
x ( le `  K ) z ) )
3122, 30impbid 191 . . . 4  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  -> 
( A. z  e.  ( Base `  K
) x ( le
`  K ) z  <-> 
x  =  .0.  )
)
3216, 31syl5bbr 259 . . 3  |-  ( ( K  e.  OP  /\  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.  ) )
3310, 32riota5 6076 . 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.  )
348, 33eqtrd 2473 1  |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713    C_ wss 3325   (/)c0 3634   class class class wbr 4289   ` cfv 5415   iota_crio 6048   Basecbs 14170   lecple 14241   lubclub 15108   0.cp0 15203   OPcops 32505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-poset 15112  df-lub 15140  df-glb 15141  df-p0 15205  df-oposet 32509
This theorem is referenced by: (None)
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