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Theorem lub0N 33863
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 2462 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2462 . . 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 3809 . . . 4  |-  (/)  C_  ( Base `  K )
76a1i 11 . . 3  |-  ( K  e.  OP  ->  (/)  C_  ( Base `  K ) )
81, 2, 3, 4, 5, 7lubval 15462 . 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 33858 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
11 ral0 3927 . . . . . . 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 2890 . . . . 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 3927 . . . . . 6  |-  A. y  e.  (/)  y ( le
`  K ) x
1514biantrur 506 . . . . 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 465 . . . . . . 7  |-  ( ( K  e.  OP  /\  x  e.  ( Base `  K ) )  ->  .0.  e.  ( Base `  K
) )
18 breq2 4446 . . . . . . . 8  |-  ( z  =  .0.  ->  (
x ( le `  K ) z  <->  x ( le `  K )  .0.  ) )
1918rspcv 3205 . . . . . . 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 33861 . . . . . 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 33860 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  z  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) z )
2423adantlr 714 . . . . . . . . 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 4445 . . . . . . . . 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 2875 . . . . 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 6264 . 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 2503 1  |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809    C_ wss 3471   (/)c0 3780   class class class wbr 4442   ` cfv 5581   iota_crio 6237   Basecbs 14481   lecple 14553   lubclub 15420   0.cp0 15515   OPcops 33846
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-poset 15424  df-lub 15452  df-glb 15453  df-p0 15517  df-oposet 33850
This theorem is referenced by: (None)
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