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Theorem lub0N 29672
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 3616 . . 3  |-  (/)  C_  ( Base `  K )
2 eqid 2404 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
4 lub0.u . . . 4  |-  .1.  =  ( lub `  K )
52, 3, 4lubval 14391 . . 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 29667 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
9 ral0 3692 . . . . . . . 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 2690 . . . . . 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 3692 . . . . . . 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 4176 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x ( le `  K ) z  <->  x ( le `  K )  .0.  ) )
1615rspcv 3008 . . . . . . . 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 29670 . . . . . . . 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 29669 . . . . . . . . . . 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 4175 . . . . . . . . . 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 2755 . . . . . 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 6535 . . 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 2436 1  |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   (/)c0 3588   class class class wbr 4172   ` cfv 5413   iota_crio 6501   Basecbs 13424   lecple 13491   lubclub 14354   0.cp0 14421   OPcops 29655
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-undef 6502  df-riota 6508  df-poset 14358  df-lub 14386  df-glb 14387  df-p0 14423  df-oposet 29659
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