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Theorem lub0N 30001
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 3496 . . 3  |-  (/)  C_  ( Base `  K )
2 eqid 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
4 lub0.u . . . 4  |-  .1.  =  ( lub `  K )
52, 3, 4lubval 14129 . . 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 29996 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
9 ral0 3571 . . . . . . . 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 2580 . . . . . 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 3571 . . . . . . 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 4043 . . . . . . . . 9  |-  ( z  =  .0.  ->  (
x ( le `  K ) z  <->  x ( le `  K )  .0.  ) )
1615rspcv 2893 . . . . . . . 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 29999 . . . . . . . 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 29998 . . . . . . . . . . 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 4042 . . . . . . . . . 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 2645 . . . . . 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 6347 . . 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 2328 1  |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   (/)c0 3468   class class class wbr 4039   ` cfv 5271   iota_crio 6313   Basecbs 13164   lecple 13231   lubclub 14092   0.cp0 14159   OPcops 29984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-glb 14125  df-p0 14161  df-oposet 29988
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