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Theorem lejoin2 15503
Description: A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
Hypotheses
Ref Expression
joinval2.b  |-  B  =  ( Base `  K
)
joinval2.l  |-  .<_  =  ( le `  K )
joinval2.j  |-  .\/  =  ( join `  K )
joinval2.k  |-  ( ph  ->  K  e.  V )
joinval2.x  |-  ( ph  ->  X  e.  B )
joinval2.y  |-  ( ph  ->  Y  e.  B )
joinlem.e  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  .\/  )
Assertion
Ref Expression
lejoin2  |-  ( ph  ->  Y  .<_  ( X  .\/  Y ) )

Proof of Theorem lejoin2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 joinval2.b . . 3  |-  B  =  ( Base `  K
)
2 joinval2.l . . 3  |-  .<_  =  ( le `  K )
3 joinval2.j . . 3  |-  .\/  =  ( join `  K )
4 joinval2.k . . 3  |-  ( ph  ->  K  e.  V )
5 joinval2.x . . 3  |-  ( ph  ->  X  e.  B )
6 joinval2.y . . 3  |-  ( ph  ->  Y  e.  B )
7 joinlem.e . . 3  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  .\/  )
81, 2, 3, 4, 5, 6, 7joinlem 15501 . 2  |-  ( ph  ->  ( ( X  .<_  ( X  .\/  Y )  /\  Y  .<_  ( X 
.\/  Y ) )  /\  A. z  e.  B  ( ( X 
.<_  z  /\  Y  .<_  z )  ->  ( X  .\/  Y )  .<_  z ) ) )
9 simplr 754 . 2  |-  ( ( ( X  .<_  ( X 
.\/  Y )  /\  Y  .<_  ( X  .\/  Y ) )  /\  A. z  e.  B  (
( X  .<_  z  /\  Y  .<_  z )  -> 
( X  .\/  Y
)  .<_  z ) )  ->  Y  .<_  ( X 
.\/  Y ) )
108, 9syl 16 1  |-  ( ph  ->  Y  .<_  ( X  .\/  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   <.cop 4033   class class class wbr 4447   dom cdm 4999   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-lub 15464  df-join 15466
This theorem is referenced by:  joinle  15504  latlej2  15551
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