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Theorem lejoin2 16211
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 16209 . 2  |-  ( ph  ->  ( ( X  .<_  ( X  .\/  Y )  /\  Y  .<_  ( X 
.\/  Y ) )  /\  A. z  e.  B  ( ( X 
.<_  z  /\  Y  .<_  z )  ->  ( X  .\/  Y )  .<_  z ) ) )
9 simplr 760 . 2  |-  ( ( ( X  .<_  ( X 
.\/  Y )  /\  Y  .<_  ( X  .\/  Y ) )  /\  A. z  e.  B  (
( X  .<_  z  /\  Y  .<_  z )  -> 
( X  .\/  Y
)  .<_  z ) )  ->  Y  .<_  ( X 
.\/  Y ) )
108, 9syl 17 1  |-  ( ph  ->  Y  .<_  ( X  .\/  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   <.cop 3999   class class class wbr 4417   dom cdm 4845   ` cfv 5592  (class class class)co 6296   Basecbs 15081   lecple 15157   joincjn 16141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-lub 16172  df-join 16174
This theorem is referenced by:  joinle  16212  latlej2  16259
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