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Theorem joinle 15770
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 15767 . . . 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 4460 . . . . . 6  |-  ( z  =  Z  ->  ( X  .<_  z  <->  X  .<_  Z ) )
12 breq2 4460 . . . . . 6  |-  ( z  =  Z  ->  ( Y  .<_  z  <->  Y  .<_  Z ) )
1311, 12anbi12d 710 . . . . 5  |-  ( z  =  Z  ->  (
( X  .<_  z  /\  Y  .<_  z )  <->  ( X  .<_  Z  /\  Y  .<_  Z ) ) )
14 breq2 4460 . . . . 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 3208 . . 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 15768 . . . 4  |-  ( ph  ->  X  .<_  ( X  .\/  Y ) )
192, 4, 5, 6, 7, 8joincl 15762 . . . . 5  |-  ( ph  ->  ( X  .\/  Y
)  e.  B )
202, 3postr 15709 . . . . 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 1230 . . . 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 15769 . . . 4  |-  ( ph  ->  Y  .<_  ( X  .\/  Y ) )
242, 3postr 15709 . . . . 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 1230 . . . 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 1395    e. wcel 1819   A.wral 2807   <.cop 4038   class class class wbr 4456   dom cdm 5008   ` cfv 5594  (class class class)co 6296   Basecbs 14643   lecple 14718   Posetcpo 15695   joincjn 15699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-poset 15701  df-lub 15730  df-join 15732
This theorem is referenced by:  latjle12  15818
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