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Theorem joinle 15189
Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.)
Hypotheses
Ref Expression
joinle.b  |-  B  =  ( Base `  K
)
joinle.l  |-  .<_  =  ( le `  K )
joinle.j  |-  .\/  =  ( join `  K )
joinle.k  |-  ( ph  ->  K  e.  Poset )
joinle.x  |-  ( ph  ->  X  e.  B )
joinle.y  |-  ( ph  ->  Y  e.  B )
joinle.z  |-  ( ph  ->  Z  e.  B )
joinle.e  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  .\/  )
Assertion
Ref Expression
joinle  |-  ( ph  ->  ( ( X  .<_  Z  /\  Y  .<_  Z )  <-> 
( X  .\/  Y
)  .<_  Z ) )

Proof of Theorem joinle
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 joinle.z . . 3  |-  ( ph  ->  Z  e.  B )
2 joinle.b . . . . 5  |-  B  =  ( Base `  K
)
3 joinle.l . . . . 5  |-  .<_  =  ( le `  K )
4 joinle.j . . . . 5  |-  .\/  =  ( join `  K )
5 joinle.k . . . . 5  |-  ( ph  ->  K  e.  Poset )
6 joinle.x . . . . 5  |-  ( ph  ->  X  e.  B )
7 joinle.y . . . . 5  |-  ( ph  ->  Y  e.  B )
8 joinle.e . . . . 5  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  .\/  )
92, 3, 4, 5, 6, 7, 8joinlem 15186 . . . 4  |-  ( ph  ->  ( ( X  .<_  ( X  .\/  Y )  /\  Y  .<_  ( X 
.\/  Y ) )  /\  A. z  e.  B  ( ( X 
.<_  z  /\  Y  .<_  z )  ->  ( X  .\/  Y )  .<_  z ) ) )
109simprd 463 . . 3  |-  ( ph  ->  A. z  e.  B  ( ( X  .<_  z  /\  Y  .<_  z )  ->  ( X  .\/  Y )  .<_  z )
)
11 breq2 4301 . . . . . 6  |-  ( z  =  Z  ->  ( X  .<_  z  <->  X  .<_  Z ) )
12 breq2 4301 . . . . . 6  |-  ( z  =  Z  ->  ( Y  .<_  z  <->  Y  .<_  Z ) )
1311, 12anbi12d 710 . . . . 5  |-  ( z  =  Z  ->  (
( X  .<_  z  /\  Y  .<_  z )  <->  ( X  .<_  Z  /\  Y  .<_  Z ) ) )
14 breq2 4301 . . . . 5  |-  ( z  =  Z  ->  (
( X  .\/  Y
)  .<_  z  <->  ( X  .\/  Y )  .<_  Z ) )
1513, 14imbi12d 320 . . . 4  |-  ( z  =  Z  ->  (
( ( X  .<_  z  /\  Y  .<_  z )  ->  ( X  .\/  Y )  .<_  z )  <->  ( ( X  .<_  Z  /\  Y  .<_  Z )  -> 
( X  .\/  Y
)  .<_  Z ) ) )
1615rspcva 3076 . . 3  |-  ( ( Z  e.  B  /\  A. z  e.  B  ( ( X  .<_  z  /\  Y  .<_  z )  -> 
( X  .\/  Y
)  .<_  z ) )  ->  ( ( X 
.<_  Z  /\  Y  .<_  Z )  ->  ( X  .\/  Y )  .<_  Z ) )
171, 10, 16syl2anc 661 . 2  |-  ( ph  ->  ( ( X  .<_  Z  /\  Y  .<_  Z )  ->  ( X  .\/  Y )  .<_  Z )
)
182, 3, 4, 5, 6, 7, 8lejoin1 15187 . . . 4  |-  ( ph  ->  X  .<_  ( X  .\/  Y ) )
192, 4, 5, 6, 7, 8joincl 15181 . . . . 5  |-  ( ph  ->  ( X  .\/  Y
)  e.  B )
202, 3postr 15128 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B ) )  -> 
( ( X  .<_  ( X  .\/  Y )  /\  ( X  .\/  Y )  .<_  Z )  ->  X  .<_  Z )
)
215, 6, 19, 1, 20syl13anc 1220 . . . 4  |-  ( ph  ->  ( ( X  .<_  ( X  .\/  Y )  /\  ( X  .\/  Y )  .<_  Z )  ->  X  .<_  Z )
)
2218, 21mpand 675 . . 3  |-  ( ph  ->  ( ( X  .\/  Y )  .<_  Z  ->  X 
.<_  Z ) )
232, 3, 4, 5, 6, 7, 8lejoin2 15188 . . . 4  |-  ( ph  ->  Y  .<_  ( X  .\/  Y ) )
242, 3postr 15128 . . . . 5  |-  ( ( K  e.  Poset  /\  ( Y  e.  B  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B ) )  -> 
( ( Y  .<_  ( X  .\/  Y )  /\  ( X  .\/  Y )  .<_  Z )  ->  Y  .<_  Z )
)
255, 7, 19, 1, 24syl13anc 1220 . . . 4  |-  ( ph  ->  ( ( Y  .<_  ( X  .\/  Y )  /\  ( X  .\/  Y )  .<_  Z )  ->  Y  .<_  Z )
)
2623, 25mpand 675 . . 3  |-  ( ph  ->  ( ( X  .\/  Y )  .<_  Z  ->  Y 
.<_  Z ) )
2722, 26jcad 533 . 2  |-  ( ph  ->  ( ( X  .\/  Y )  .<_  Z  ->  ( X  .<_  Z  /\  Y  .<_  Z ) ) )
2817, 27impbid 191 1  |-  ( ph  ->  ( ( X  .<_  Z  /\  Y  .<_  Z )  <-> 
( X  .\/  Y
)  .<_  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   <.cop 3888   class class class wbr 4297   dom cdm 4845   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   Posetcpo 15115   joincjn 15119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-poset 15121  df-lub 15149  df-join 15151
This theorem is referenced by:  latjle12  15237
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