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Theorem omlim 6527
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
Dummy variable  y is distinct from all other variables.
Allowed substitution hint:    C( x)

Proof of Theorem omlim
StepHypRef Expression
1 limelon 4454 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  B  e.  On )
2 simpr 449 . . 3  |-  ( ( B  e.  C  /\  Lim  B )  ->  Lim  B )
31, 2jca 520 . 2  |-  ( ( B  e.  C  /\  Lim  B )  ->  ( B  e.  On  /\  Lim  B ) )
4 rdglim2a 6441 . . . 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 454 . . 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 6506 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  B ) )
7 onelon 4416 . . . . . . . 8  |-  ( ( B  e.  On  /\  x  e.  B )  ->  x  e.  On )
8 omv 6506 . . . . . . . 8  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  .o  x
)  =  ( rec ( ( y  e. 
_V  |->  ( y  +o  A ) ) ,  (/) ) `  x ) )
97, 8sylan2 462 . . . . . . 7  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  x  e.  B ) )  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
109anassrs 631 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  x  e.  B
)  ->  ( A  .o  x )  =  ( rec ( ( y  e.  _V  |->  ( y  +o  A ) ) ,  (/) ) `  x
) )
1110iuneq2dv 3927 . . . . 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 2298 . . . 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 699 . . 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 225 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\ 
Lim  B ) )  ->  ( A  .o  B )  =  U_ x  e.  B  ( A  .o  x ) )
153, 14sylan2 462 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 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2789   (/)c0 3456   U_ciun 3906    e. cmpt 4078   Oncon0 4391   Lim wlim 4392   ` cfv 5221  (class class class)co 5819   reccrdg 6417    +o coa 6471    .o comu 6472
This theorem is referenced by:  omcl  6530  om0r  6533  om1r  6536  omordi  6559  omwordri  6565  omordlim  6570  omlimcl  6571  odi  6572  omass  6573  omeulem1  6575  oeoalem  6589  oeoelem  6591  omabslem  6639  omabs  6640
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-recs 6383  df-rdg 6418  df-omul 6479
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