MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omlim Unicode version

Theorem omlim 6619
Description: Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
omlim  |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem omlim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 limelon 4537 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 447 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 518 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 6533 . . . 4  |-  ( ( B  e.  On  /\  Lim  B )  ->  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B )  =  U_ x  e.  B  ( rec (
( y  e.  _V  |->  ( y  +o  A
) ) ,  (/) ) `  x )
)
54adantl 452 . . 3  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( rec (
( y  e.  _V  |->  ( y  +o  A
) ) ,  (/) ) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
6 omv 6598 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B ) )
7 onelon 4499 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 omv 6598 . . . . . . . 8  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  .o  x
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  x ) )
97, 8sylan2 460 . . . . . . 7  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
109anassrs 629 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
1110iuneq2dv 4007 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  U_ x  e.  B  ( A  .o  x
)  =  U_ x  e.  B  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  x ) )
126, 11eqeq12d 2372 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x )  <->  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B )  =  U_ x  e.  B  ( rec (
( y  e.  _V  |->  ( y  +o  A
) ) ,  (/) ) `  x )
) )
1312adantrr 697 . . 3  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( ( A  .o  B )  = 
U_ x  e.  B  ( A  .o  x
)  <->  ( rec (
( y  e.  _V  |->  ( y  +o  A
) ) ,  (/) ) `  B )  =  U_ x  e.  B  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) ) )
145, 13mpbird 223 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
153, 14sylan2 460 1  |-  ( ( A  e.  On  /\  ( B  e.  C  /\  Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531   U_ciun 3986    e. cmpt 4158   Oncon0 4474   Lim wlim 4475   ` cfv 5337  (class class class)co 5945   reccrdg 6509    +o coa 6563    .o comu 6564
This theorem is referenced by:  omcl  6622  om0r  6625  om1r  6628  omordi  6651  omwordri  6657  omordlim  6662  omlimcl  6663  odi  6664  omass  6665  omeulem1  6667  oeoalem  6681  oeoelem  6683  omabslem  6731  omabs  6732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-recs 6475  df-rdg 6510  df-omul 6571
  Copyright terms: Public domain W3C validator